Ex Parte Perronnin et al

Patent Trial and Appeal Board

Feb 23, 2017

12109496 (P.T.A.B. Feb. 23, 2017)

UNITED STATES PATENT AND TRADEMARK OFFICE
UNITED STATES DEPARTMENT OF COMMERCE
United States Patent and Trademark Office
Address: COMMISSIONER FOR PATENTS
P.O. Box 1450
Alexandria, Virginia 22313-1450
www.uspto.gov
APPLICATION NO. FILING DATE FIRST NAMED INVENTOR ATTORNEY DOCKET NO. CONFIRMATION NO.
12/109,496 04/25/2008 Florent Perronnin 20070981USNP-XER1867US01 3575
62095 7590 02/24/2017
FAY SHARPE / XEROX - ROCHESTER
1228 EUCLID AVENUE, 5TH FLOOR
THE HALLE BUILDING
CLEVELAND, OH 44115
EXAMINER
ROSTAMI, MOHAMMAD S
ART UNIT PAPER NUMBER
2154
MAIL DATE DELIVERY MODE
02/24/2017 PAPER
Please find below and/or attached an Office communication concerning this application or proceeding.
The time period for reply, if any, is set in the attached communication.
PTOL-90A (Rev. 04/07)
UNITED STATES PATENT AND TRADEMARK OFFICE
____________________

BEFORE THE PATENT TRIAL AND APPEAL BOARD
____________________

Ex parte FLORENT PERRONNIN and GUILLAUME BOUCHARD
____________________

Appeal 2016-006538
Application 12/109,496
Technology Center 2100
____________________

Before ERIC S. FRAHM, LARRY J. HUME, and
CATHERINE SHIANG, Administrative Patent Judges.

FRAHM, Administrative Patent Judge.

DECISION ON APPEAL

STATEMENT OF THE CASE
Introduction
Appellants appeal under 35 U.S.C. § 134(a) from a Final Rejection of
claims 1–21 and 23, all the claims pending in the application. We have
jurisdiction under 35 U.S.C. § 6(b).
We affirm-in-part.
Appellants’ Disclosed Invention
Appellants disclose a system and method of clustering for use in data
storage and information management, whereby the clustering of objects uses
a nonnegative sparse similarity matrix and various combinations of
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Application 12/109,496

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mathematical calculations to factorize the matrices and allocate objects
based on factor matrices generated by factorization of the nonnegative
sparse similarity matrix (Spec. ¶¶ 1–3; Title; Abs.; The Figure).
Exemplary Claims
An understanding of the invention can be derived from a reading of
exemplary claims 1, 3, and 8, which are reproduced below with emphases
added to contested limitations:
1. A clustering method comprising:
constructing a nonnegative sparse similarity matrix for a
set of objects, the constructing including one of:
constructing an e graph defining the nonnegative sparse
similarity matrix, the É› graph including nodes for object
pairs conditional upon a similarity measure of the object pair
exceeding a threshold É›,
constructing a K-nearest neighbors (K-NN) directed
graph defining the nonnegative sparse similarity matrix, the K-
NN directed graph including a node for first and second objects
of the set of objects conditional upon the second object being one
of K nearest neighbors of the first object, and
constructing an adjacency matrix having matrix elements
corresponding to object pairs, the matrix element values
being nonnegative values indicative of similarity of the
corresponding object pairs, and deriving a commute time matrix
from the adjacency matrix;
performing nonnegative factorization of the nonnegative
sparse similarity matrix; and
allocating objects of the set of objects to clusters based on
factor matrices generated by the nonnegative factorization of the
nonnegative sparse similarity matrix;
wherein the constructing, performing, and allocating are
performed by a processor executing software or firmware.
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3. The clustering method as set forth in claim 1, wherein the
constructing comprises:
constructing an É› graph defining the nonnegative sparse
similarity matrix, the É› graph including nodes for object pairs
conditional upon a similarity measure of the object pair
exceeding a threshold É› .

8. The clustering method as set forth in claim 1, wherein the
performing comprises:
optimizing factor matrices A and B to minimize a
Frobenius norm between the nonnegative sparse similarity
matrix and a matrix product AxB.

The Examiner’s Rejections
(1) The Examiner rejected claims 1, 3, 4, and 7 under 35 U.S.C.
§ 112(d) as failing to further limit the subject matter claimed, i.e., the subject
matter recited in independent claim 1. Final Act. 4–6.
(2) The Examiner rejected claims 1–5, 8, 11, 16–18, 20, 21, and
23–29 under 35 U.S.C. § 103(a) as being unpatentable over the combination
of Geshwind (US 2006/0155751 A1; issued Jul. 13, 2006) and Koren (US
2009/0083258 A1; published Mar. 26, 2009). Final Act. 6–19.
(3) The Examiner rejected claim 7 under 35 U.S.C. § 103(a) as
being unpatentable over the combination of Geshwind, Koren, and Rifkin
(US 2006/0235812 A1; published Oct. 19, 2006). Final Act. 19–21.
(4) The Examiner rejected claims 9, 10, and 12–14 under 35 U.S.C.
§ 103(a) as being unpatentable over the combination of Geshwind, Koren,
and Tamayo (US 2005/0246354 A1; published Nov. 3, 2005). Final Act.
21–24.
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Issues on Appeal1
Based on Appellants’ arguments in the Appeal Brief (App. Br. 5–23)
and the Reply Brief (2–14), in light of the Examiner’s response to
Appellants’ arguments in the Appeal Brief (Ans. 4–14), the following three
principal issues are presented on appeal:
(1) Did the Examiner err in rejecting claims 1, 3, 4, and 7 under
35 U.S.C. § 112(d) as failing to further limit the subject matter claimed, i.e.,
the subject matter recited in independent claim 1?
(2) Did the Examiner err in rejecting claims 1–5, 7–12, 15–18, 20,
21, and 23–29 because the combination (i) is improper, and/or (ii) fails to
teach or suggest the limitations at issue in representative independent claim
1?
(3) Did the Examiner err in rejecting claim 8 because the Official
Notice that a Frobenius norm is equivalent to, or a synonym of, a Hilbert-
Schmidt norm, was improperly taken or fails to support the proposition
relied upon as teaching?

1 Independent claims 1, 11, and 23 (and claims 2 –5, 8, 16–18, 20, 21, and
24–29 which depend respectively therefrom) contain the same disputed
limitations pertaining to clustering by constructing a nonnegative sparse
similarity matrix. Appellants do not present separate arguments for claims
2–5, 16–18, 20, 21, and 24–29, and rely on the arguments presented as to
claims 1, 11, and 23 (see App. Br. 8–19). We select claim 1 as
representative of claims 1–5, 11, 16–18, 20, 21, and 23–29. Claim 8,
separately argued, will be discussed separately. Because (i) claims 7, 9, 10,
and 12 each ultimately depend from representative independent claim 1; and
(ii) Appellants rely on the arguments presented for claim 1 as to the
patentability of claims 7, 9, 10, and 12, the outcome for the obviousness
rejections of claims 7, 9, 10, and 12 will stand/fall with the outcome as to
representative claim 1.
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(4) Did the Examiner err in rejecting claims 13 and 14 for obviousness
over Geshwind, Koren, and Tamayo, because none of the applied references,
taken individually or in combination, disclose initializing as recited?

ANALYSIS
We have reviewed the Examiner’s rejections (Final Act. 4–24) in light
of Appellants’ contentions in the Appeal Brief (App. Br. 5–23) and the
Reply Brief (Reply Br. 2–14) that the Examiner has erred, as well as the
Examiner’s response to Appellants’ arguments in the Appeal Brief (Ans. 3–
14). We provide the following for emphasis with regard to each of the four
issues before us on appeal.
Rejection of Claims 1, 3, 4, and 7 Under 35 U.S.C. § 112(d)
At the outset, we note our agreement with Appellants (App. Br. 8) that
claim 1 is independent, and therefore is not properly rejected as failing to
further limit itself. Accordingly, we do not sustain the Examiner’s rejection
of claim 1. Now, we turn to the issue of whether or not claims 3, 4, and 7
further limit claim 1.
It is axiomatic that a dependent claim cannot be broader than the
claim from which it depends. See 35 U.S.C. § 112(d) (“[A] claim in
dependent form shall [1] contain a reference to a claim previously set forth
and then [2] specify a further limitation of the subject matter claimed.”); see
also Intamin Ltd. v. Magnetar Techs., Corp., 483 F.3d 1328, 1335 (Fed. Cir.
2007) (“An independent claim impliedly embraces more subject matter than
its narrower dependent claim.”); AK Steel Corp. v. Sollac & Ugine, 344 F.3d
1234, 1242 (Fed. Cir. 2003) (“Under the doctrine of claim differentiation,
dependent claims are presumed to be of narrower scope than the independent
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Application 12/109,496

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claims from which they depend.”); Pfizer, Inc. v. Ranbaxy Labs. Ltd., 457
F.3d 1284, 1292 (Fed. Cir. 2006).
In the instant case, claims 3, 4, and 7 recite a specific type of
construction operation as being the construction operation used as the “one
of” constructions used in the method of claim 1, thus claim 1 covers that one
type of construction. Furthermore, because a dependent claim narrows the
claim from which it depends, it must “incorporate . . . all the limitations of
the claim to which it refers.” 35 U.S.C. § 112(d).2 In the instant case,
claims 3, 4, and 7 must incorporate all the limitations of claim 1, and the
construction used in claim 1, when modified by any of claims 3, 4, and 7,
must be a specific one of the three listed in claim 1. Because claim 1 only
requires one of the constructions of the three listed in claim 1, and claims 3,
4, and 7 each recite a specific single type of construction to be used from the
group of three possible constructions listed in claim 1, claims 3, 4, and 7
further limit claim 1.
In view of the foregoing, we agree with Appellants’ arguments (App.
Br. 8–9) that claims 3, 4, and 7 are narrower than claim 1. Accordingly, we
do not sustain the Examiner’s rejection of claims 3, 4, and 7 under 35 U.S.C.
§ 112(d).

2 “One or more claims may be presented in dependent form, referring back
to and further limiting another claim or claims in the same application.” 37
C.F.R. § 1.75(c); see also MPEP §§ 608.01(i), (n). “Claims in dependent
form shall be construed to include all the limitations of the claim
incorporated by reference into the dependent claim.” 37 C.F.R. § 1.75(c);
see also MPEP §§ 608.01(i), (n).

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Obviousness Rejections of Claims 1–5, 7–12, 15–18, 20, 21, and 23–29
We disagree with the Appellants’ contentions as to representative
independent claim 1 and dependent claim 8, separately argued. With regard
to claim 1 and 8, we adopt as our own (1) the findings and reasons set forth
by the Examiner in the action from which this appeal is taken (Final Act. 6–
10), and (2) the reasons set forth by the Examiner in the Examiner’s Answer
in response to the Appellants’ Appeal Brief (Ans. 4–13). We provide the
following comments concerning Appellants’ arguments and certain
teachings and suggestions of the references as follows.
We note that each reference cited by the Examiner must be read, not
in isolation, but for what it fairly teaches in combination with the prior art as
a whole. See In re Merck & Co., 800 F.2d 1091, 1097 (Fed. Cir. 1986) (one
cannot show non-obviousness by attacking references individually where the
rejections are based on combinations of references). In this light,
Appellants’ arguments as to representative independent claim 1 (App. Br.
10–15) concerning the individual shortcomings in the teachings of Geshwind
and Koren are not persuasive, and are not convincing of the non-obviousness
of the claimed invention set forth in representative independent claim 1.
With regard to representative independent claim 1, the Examiner has
relied upon the combination of Geshwind and Koren as teaching or
suggesting the clustering method to construct a nonnegative sparse similarity
matrix for a set of objects using one of the construction operations recited in
claim 1. The Examiner relies upon Geshwind as disclosing clustering and
construction of a nonnegative sparse similarity matrix (see Final Act. 6–9);
and relies upon Koren as disclosing clustering, factorization of a matrix,
using a nonnegative sparse similarity matrix for objects, and allocating
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objects to matrices using the sparse similarity matrix (see Final Act. 9; Ans.
4–8 citing Koren ¶¶ 78, 79, and 83). We agree with the Examiner’s findings
regarding Geshwind and Koren.
We also agree with the Examiner’s conclusion that the combination is
properly made (see Final Act. 9–10; Ans. 8–10) and the combination teaches
and/or suggests the recited limitations of representative independent claim 1.
Specifically, we agree with the Examiner’s determination that
because neighborhood based methods are intuitive and relatively
simple to implement, without a need to present many parameters
or to conduct an extensive training stage. They also allow for
presenting a user with similar items that he or she has rated, and
giving the user an opportunity to change previous ratings in
accordance with his or her present tastes, with the understanding
that this will affect subsequent ratings.
Ans. 9. We also agree with the Examiner’s determination that it would have
been obvious to combine the teachings of Geshwind and Koren “because
Koren’s system would have allowed Geshwind to facilitate a system for
performing nonnegative factorization of the nonnegative sparse similarity
matrix; allocating objects of the set of objects to clusters based on factor
matrices generated by the nonnegative factorization of the nonnegative
sparse similarity matrix” (Final Act. 9–10).
In addition, the portions of Koren cited by the Examiner (see Ans. 4–
5) strongly suggest that modifying Geshwind with the teachings of Koren
would provide the benefits of (i) improving prediction accuracy (Koren
¶ 90); and (ii) alleviating computational complexity (Koren ¶ 86).
In view of the foregoing, we sustain the Examiner’s obviousness
rejection of representative independent claim 1, as well as claims 2–5, 8, 11,
16–18, 20, 21, and 23–29 grouped therewith.
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We also sustain the Examiner’s obviousness rejections of (i) claim 7
over the combination of Geshwind, Koren, and Rifkin; and (ii) claims 9, 10,
and 12 over the combination of Geshwind, Koren, and Tamayo for the same
reasons as provided as to claim 1.
Obviousness Rejection of Claim 8: Official Notice
With regard to the obviousness rejection of claim 8, the Examiner
relies upon Official Notice that it is well-known to those of ordinary skill in
the art that a Frobenius norm is equivalent to, or a synonym of, a Hilbert-
Schmidt norm (Ans. 10). The Examiner is correct, as evidenced by:
(1) James E. Gentle, Matrix Algebra: Theory, Computations, and
Applications in Statistics, p. 132 (2007):3 “The Frobenius norm is also often
called the “usual norm”, which emphasizes the fact that it is one of the most
useful matrix norms. Other names sometimes used to refer to the Frobenius
norm are Hilbert-Schmidt norm and Schur norm;” and
(2) Carl D. Meyer, Matrix Analysis and Applied Linear Algebra,
Chapter 5.2, Matrix Norms, p. 279 (2000):4 “This is one of the simplest
notions of a matrix norm, and it is called the Frobenius (p. 662) norm (older
texts refer to it as the Hilbert–Schmidt norm or the Schur norm).”
Copies of these two documents are attached to this Decision on a form
PTO-892. In view of the foregoing, Appellants’ arguments (App. Br. 10;
Reply Br. 8) that Gilbert merely discloses a Hilbert-Schmidt norm and fails

3 See also http://saba.kntu.ac.ir/eecd/sedghizadeh/Ebooks/
Matrix_Analysis.pdf, last viewed on February 20, 2017.
4 See also http://www.matrixanalysis.com/page279.pdf, last viewed on
February 20, 2017.
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to disclose the equivalent of the recited Frobenius norm recited in claim 8
are unpersuasive.
For this reason, we sustain the Examiner’s obviousness rejection of
claim 8.
Obviousness Rejection of Claims 13 and 14
We agree with Appellants’ contentions as to claims 13 and 14 (App.
Br. 20–21; Reply Br. 12–13) that none of the applied references Geshwind,
Koren, and Tamayo teach or suggest initiation as recited, and Tamayo
teaches random initialization. Accordingly, we do not sustain the
Examiner’s obviousness rejection of claims 13 and 14.

CONCLUSIONS
(1) Appellants have shown the Examiner erred in rejecting claims
1, 3, 4, and 7 under 35 U.S.C. § 112(d) as failing to further limit the subject
matter claimed, i.e., the subject matter recited in independent claim 1.
(2) The Examiner did not err in rejecting claims 1–5, 7–12, 15–18,
20, 21, and 23–29 under 35 U.S.C. § 103(a) because the combination (i) is
proper, and (ii) teaches or suggests the limitations at issue in representative
independent claim 1.
(3) The Examiner did not err in rejecting claim 8 under 35 U.S.C.
§ 103(a) because the Official Notice that a Frobenius norm is equivalent to,
or a synonym of, a Hilbert-Schmidt norm, was properly taken, as evidenced
by the documents cited to Appellants in this Decision (see supra
accompanying form PTO-892 and Evidence Appendix).
(4) Appellants have established the Examiner erred in rejecting claims
13 and 14 as being unpatentable under 35 U.S.C. § 103(a) because none of
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Application 12/109,496

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the applied references, taken singly or in combination, disclose initialization
as recited in claims 13 and/or 14.

DECISION
We affirm the Examiner’s rejections of claims 1–5, 7–12, 15–18, 20,
21, and 23–29 under 35 U.S.C. § 103(a).
We reverse the Examiner’s rejections of (i) claims 1, 3, 4, and 7 under
35 U.S.C. 112(d); and (ii) claims 13 and 14 under 35 U.S.C. § 103(a).
No time period for taking any subsequent action in connection with
this appeal may be extended under 37 C.F.R. § 1.136(a)(1)(iv).

AFFIRMED-IN-PART

Appeal 2016-006538
Application 12/109,496

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EVIDENCE APPENDIX

James E. Gentle, Matrix Algebra: Theory, Computations, and
Applications in Statistics, p. 132 (2007).

Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, Chapter
5.2, Matrix Norms, p. 279 (2000).

Notice of References Cited

Application/Control No.

12/109,496
Applicant(s)/Patent Under Patent
Appeal No.
2016-006538
Administrative Patent Judge 1
Eric S. Frahm
Art Unit
2100

Page 1 of 1

U.S. PATENT DOCUMENTS

* Document Number Country Code-Number-Kind Code Date MM-YYYY

Name

Classification

A US-

B US-

C US-

D US-

E US-

F US-

G US-

H US-

I US-

J US-

K US-

L US-

M US-
FOREIGN PATENT DOCUMENTS

* Document Number Country Code-Number-Kind Code Date MM-YYYY

Country

Name

Classification

N

O

P

Q

R

S

T
NON-PATENT DOCUMENTS
* Include as applicable: Author, Title Date, Publisher, Edition or Volume, Pertinent Pages)

U

James E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, p. 132 (2007)

V

Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, Chapter 5.2, Matrix Norms, p. 279
(2000).

W

X

*A copy of this reference is being furnished with the associated Board decision from this appeal..
Dates in MM-YYYY format are publication dates. Classifications may be US or foreign.

U.S. Patent and Trademark Office
PTO-892 (Rev. 01-2001) Notice of References Cited Part of Paper No.
James E. Gentle
Matrix Algebra
Theory, Computations, and Applications
in Statistics
Editorial Board
George Casella Stephen Fienberg Ingram Olkin
Department of Statistics Department of Statistics
University of Florida Carnegie Mellon University Stanford University
Gainesville, FL 32611-8545 Pittsburgh, PA 15213-3890 Stanford, CA 94305
USA USA USA
Printed on acid-free paper.
© 2007 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY, 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or
not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 1
springer.com
James E. Gentle
Department of Computational
and Data Sciences
George Mason University
4400 University Drive
Fairfax, VA 22030-4444
jgentle@gmu.edu
ISBN :978-0-387-70872-0 e-ISBN :978-0-387-70873-7
Department of Statistics
Library of Congress Control Number: 2007930269
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Part I Linear Algebra
1 Basic Vector/Matrix Structure and Notation . . . . . . . . . . . . . . 3
1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Vectors and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Operations on Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Linear Combinations and Linear Independence . . . . . . . . 10
2.1.2 Vector Spaces and Spaces of Vectors . . . . . . . . . . . . . . . . . 11
2.1.3 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.4 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.5 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.6 Normalized Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.7 Metrics and Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.8 Orthogonal Vectors and Orthogonal Vector Spaces . . . . . 22
2.1.9 The “One Vector” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Cartesian Coordinates and Geometrical Properties of Vectors . 24
2.2.1 Cartesian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Angles between Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 Orthogonalization Transformations . . . . . . . . . . . . . . . . . . 27
2.2.5 Orthonormal Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.6 Approximation of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.7 Flats, Affine Spaces, and Hyperplanes . . . . . . . . . . . . . . . . 31
2.2.8 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
xvi Contents
2.2.9 Cross Products in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Centered Vectors and Variances and Covariances of Vectors . . . 33
2.3.1 The Mean and Centered Vectors . . . . . . . . . . . . . . . . . . . . 34
2.3.2 The Standard Deviation, the Variance,
and Scaled Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Covariances and Correlations between Vectors . . . . . . . . 36
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Basic Properties of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Matrix Shaping Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.3 Matrix Addition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.4 Scalar-Valued Operators on Square Matrices:
The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.5 Scalar-Valued Operators on Square Matrices:
The Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Multiplication of Matrices and Multiplication
of Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Matrix Multiplication (Cayley) . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Multiplication of Partitioned Matrices . . . . . . . . . . . . . . . . 61
3.2.3 Elementary Operations on Matrices . . . . . . . . . . . . . . . . . . 61
3.2.4 Traces and Determinants of Square Cayley Products . . . 67
3.2.5 Multiplication of Matrices and Vectors . . . . . . . . . . . . . . . 68
3.2.6 Outer Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.7 Bilinear and Quadratic Forms; Definiteness . . . . . . . . . . . 69
3.2.8 Anisometric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.9 Other Kinds of Matrix Multiplication . . . . . . . . . . . . . . . . 72
3.3 Matrix Rank and the Inverse of a Full Rank Matrix . . . . . . . . . . 76
3.3.1 The Rank of Partitioned Matrices, Products
of Matrices, and Sums of Matrices . . . . . . . . . . . . . . . . . . . 78
3.3.2 Full Rank Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.3 Full Rank Matrices and Matrix Inverses . . . . . . . . . . . . . . 81
3.3.4 Full Rank Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.5 Equivalent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.6 Multiplication by Full Rank Matrices . . . . . . . . . . . . . . . . 88
3.3.7 Products of the Form ATA . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.8 A Lower Bound on the Rank of a Matrix Product . . . . . 92
3.3.9 Determinants of Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.10 Inverses of Products and Sums of Matrices . . . . . . . . . . . 93
3.3.11 Inverses of Matrices with Special Forms . . . . . . . . . . . . . . 94
3.3.12 Determining the Rank of a Matrix . . . . . . . . . . . . . . . . . . . 94
3.4 More on Partitioned Square Matrices: The Schur Complement 95
3.4.1 Inverses of Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . 95
3.4.2 Determinants of Partitioned Matrices . . . . . . . . . . . . . . . . 96
Contents xvii
3.5 Linear Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.1 Solutions of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5.2 Null Space: The Orthogonal Complement . . . . . . . . . . . . . 99
3.6 Generalized Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.1 Generalized Inverses of Sums of Matrices . . . . . . . . . . . . . 101
3.6.2 Generalized Inverses of Partitioned Matrices . . . . . . . . . . 101
3.6.3 Pseudoinverse or Moore-Penrose Inverse . . . . . . . . . . . . . . 101
3.7 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.8 Eigenanalysis; Canonical Factorizations . . . . . . . . . . . . . . . . . . . . 105
3.8.1 Basic Properties of Eigenvalues and Eigenvectors . . . . . . 107
3.8.2 The Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . . . 108
3.8.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.8.4 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.8.5 Similar Canonical Factorization;
Diagonalizable Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.8.6 Properties of Diagonalizable Matrices . . . . . . . . . . . . . . . . 118
3.8.7 Eigenanalysis of Symmetric Matrices . . . . . . . . . . . . . . . . . 119
3.8.8 Positive Definite and Nonnegative Definite Matrices . . . 124
3.8.9 The Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . 126
3.8.10 Singular Values and the Singular Value Decomposition . 127
3.9 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.9.1 Matrix Norms Induced from Vector Norms . . . . . . . . . . . 129
3.9.2 The Frobenius Norm — The “Usual” Norm . . . . . . . . . . . 131
3.9.3 Matrix Norm Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.9.4 The Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.9.5 Convergence of a Matrix Power Series . . . . . . . . . . . . . . . . 134
3.10 Approximation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4 Vector/Matrix Derivatives and Integrals . . . . . . . . . . . . . . . . . . . 145
4.1 Basics of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2 Types of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2.1 Differentiation with Respect to a Scalar . . . . . . . . . . . . . . 149
4.2.2 Differentiation with Respect to a Vector . . . . . . . . . . . . . . 150
4.2.3 Differentiation with Respect to a Matrix . . . . . . . . . . . . . 154
4.3 Optimization of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.3.1 Stationary Points of Functions . . . . . . . . . . . . . . . . . . . . . . 156
4.3.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.3.3 Optimization of Functions with Restrictions . . . . . . . . . . 159
4.4 Multiparameter Likelihood Functions . . . . . . . . . . . . . . . . . . . . . . 163
4.5 Integration and Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.5.1 Multidimensional Integrals and Integrals Involving
Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.5.2 Integration Combined with Other Operations . . . . . . . . . 166
4.5.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
xviii Contents
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5 Matrix Transformations and Factorizations . . . . . . . . . . . . . . . . 173
5.1 Transformations by Orthogonal Matrices . . . . . . . . . . . . . . . . . . . 174
5.2 Geometric Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.2.2 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.2.3 Translations; Homogeneous Coordinates . . . . . . . . . . . . . . 178
5.3 Householder Transformations (Reflections) . . . . . . . . . . . . . . . . . . 180
5.4 Givens Transformations (Rotations) . . . . . . . . . . . . . . . . . . . . . . . 182
5.5 Factorization of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.6 LU and LDU Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.7 QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.7.1 Householder Reflections to Form the QR Factorization . 190
5.7.2 Givens Rotations to Form the QR Factorization . . . . . . . 192
5.7.3 Gram-Schmidt Transformations to Form the
QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.8 Singular Value Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.9 Factorizations of Nonnegative Definite Matrices . . . . . . . . . . . . . 193
5.9.1 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.9.2 Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.9.3 Factorizations of a Gramian Matrix . . . . . . . . . . . . . . . . . . 196
5.10 Incomplete Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6 Solution of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.1 Condition of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Direct Methods for Consistent Systems . . . . . . . . . . . . . . . . . . . . . 206
6.2.1 Gaussian Elimination and Matrix Factorizations . . . . . . . 207
6.2.2 Choice of Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.3 Iterative Methods for Consistent Systems . . . . . . . . . . . . . . . . . . . 211
6.3.1 The Gauss-Seidel Method with
Successive Overrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.3.2 Conjugate Gradient Methods for Symmetric
Positive Definite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.3.3 Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4 Numerical Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.5 Iterative Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.6 Updating a Solution to a Consistent System . . . . . . . . . . . . . . . . 220
6.7 Overdetermined Systems; Least Squares . . . . . . . . . . . . . . . . . . . . 222
6.7.1 Least Squares Solution of an Overdetermined System . . 224
6.7.2 Least Squares with a Full Rank Coefficient Matrix . . . . . 226
6.7.3 Least Squares with a Coefficient Matrix
Not of Full Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Contents xix
6.7.4 Updating a Least Squares Solution
of an Overdetermined System . . . . . . . . . . . . . . . . . . . . . . . 228
6.8 Other Solutions of Overdetermined Systems. . . . . . . . . . . . . . . . . 229
6.8.1 Solutions that Minimize Other Norms of the Residuals . 230
6.8.2 Regularized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.8.3 Minimizing Orthogonal Distances . . . . . . . . . . . . . . . . . . . . 234
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7 Evaluation of Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 241
7.1 General Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.1.1 Eigenvalues from Eigenvectors and Vice Versa . . . . . . . . . 242
7.1.2 Deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.1.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.2 Power Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.3 Jacobi Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.4 QR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.5 Krylov Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.6 Generalized Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.7 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Part II Applications in Data Analysis
8 Special Matrices and Operations Useful in Modeling
and Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.1 Data Matrices and Association Matrices . . . . . . . . . . . . . . . . . . . . 261
8.1.1 Flat Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.1.2 Graphs and Other Data Structures . . . . . . . . . . . . . . . . . . 262
8.1.3 Probability Distribution Models . . . . . . . . . . . . . . . . . . . . . 269
8.1.4 Association Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.2 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.3 Nonnegative Definite Matrices; Cholesky Factorization . . . . . . . 275
8.4 Positive Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.5 Idempotent and Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . 280
8.5.1 Idempotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
8.5.2 Projection Matrices: Symmetric Idempotent Matrices . . 286
8.6 Special Matrices Occurring in Data Analysis . . . . . . . . . . . . . . . . 287
8.6.1 Gramian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.6.2 Projection and Smoothing Matrices . . . . . . . . . . . . . . . . . . 290
8.6.3 Centered Matrices and Variance-Covariance Matrices . . 293
8.6.4 The Generalized Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 296
8.6.5 Similarity Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
8.6.6 Dissimilarity Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
8.7 Nonnegative and Positive Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 299
xx Contents
8.7.1 Properties of Square Positive Matrices . . . . . . . . . . . . . . . 301
8.7.2 Irreducible Square Nonnegative Matrices . . . . . . . . . . . . . 302
8.7.3 Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
8.7.4 Leslie Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
8.8 Other Matrices with Special Structures . . . . . . . . . . . . . . . . . . . . . 307
8.8.1 Helmert Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.8.2 Vandermonde Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.8.3 Hadamard Matrices and Orthogonal Arrays . . . . . . . . . . . 310
8.8.4 Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.8.5 Hankel Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
8.8.6 Cauchy Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
8.8.7 Matrices Useful in Graph Theory . . . . . . . . . . . . . . . . . . . . 313
8.8.8 M -Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
9 Selected Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 321
9.1 Multivariate Probability Distributions . . . . . . . . . . . . . . . . . . . . . . 322
9.1.1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . 322
9.1.2 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . 323
9.1.3 Derived Distributions and Cochran’s Theorem . . . . . . . . 323
9.2 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
9.2.1 Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
9.2.2 Linear Models and Least Squares . . . . . . . . . . . . . . . . . . . . 330
9.2.3 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
9.2.4 The Normal Equations and the Sweep Operator . . . . . . . 335
9.2.5 Linear Least Squares Subject to Linear
Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
9.2.6 Weighted Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
9.2.7 Updating Linear Regression Statistics . . . . . . . . . . . . . . . . 338
9.2.8 Linear Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.3 Principal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.3.1 Principal Components of a Random Vector . . . . . . . . . . . 342
9.3.2 Principal Components of Data . . . . . . . . . . . . . . . . . . . . . . 343
9.4 Condition of Models and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.4.1 Ill-Conditioning in Statistical Applications . . . . . . . . . . . . 346
9.4.2 Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
9.4.3 Principal Components Regression . . . . . . . . . . . . . . . . . . . 348
9.4.4 Shrinkage Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
9.4.5 Testing the Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 350
9.4.6 Incomplete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
9.5 Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.6 Multivariate Random Number Generation . . . . . . . . . . . . . . . . . . 358
9.7 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9.7.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9.7.2 Markovian Population Models . . . . . . . . . . . . . . . . . . . . . . . 362
Contents xxi
9.7.3 Autoregressive Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Part III Numerical Methods and Software
10 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
10.1 Digital Representation of Numeric Data . . . . . . . . . . . . . . . . . . . . 377
10.1.1 The Fixed-Point Number System . . . . . . . . . . . . . . . . . . . . 378
10.1.2 The Floating-Point Model for Real Numbers . . . . . . . . . . 379
10.1.3 Language Constructs for Representing Numeric Data . . 386
10.1.4 Other Variations in the Representation of Data;
Portability of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
10.2 Computer Operations on Numeric Data . . . . . . . . . . . . . . . . . . . . 393
10.2.1 Fixed-Point Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
10.2.2 Floating-Point Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 395
10.2.3 Exact Computations; Rational Fractions . . . . . . . . . . . . . 399
10.2.4 Language Constructs for Operations
on Numeric Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
10.3 Numerical Algorithms and Analysis . . . . . . . . . . . . . . . . . . . . . . . . 403
10.3.1 Error in Numerical Computations . . . . . . . . . . . . . . . . . . . 404
10.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
10.3.3 Iterations and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 417
10.3.4 Other Computational Techniques . . . . . . . . . . . . . . . . . . . . 419
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
11 Numerical Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
11.1 Computer Representation of Vectors and Matrices . . . . . . . . . . . 429
11.2 General Computational Considerations
for Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
11.2.1 Relative Magnitudes of Operands . . . . . . . . . . . . . . . . . . . . 431
11.2.2 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
11.2.3 Assessing Computational Errors . . . . . . . . . . . . . . . . . . . . . 434
11.3 Multiplication of Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . 435
11.4 Other Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
12 Software for Numerical Linear Algebra . . . . . . . . . . . . . . . . . . . . 445
12.1 Fortran and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
12.1.1 Programming Considerations . . . . . . . . . . . . . . . . . . . . . . . 448
12.1.2 Fortran 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
12.1.3 Matrix and Vector Classes in C++ . . . . . . . . . . . . . . . . . . 453
12.1.4 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
12.1.5 The IMSLTM Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
12.1.6 Libraries for Parallel Processing . . . . . . . . . . . . . . . . . . . . . 460
xxii Contents
12.2 Interactive Systems for Array Manipulation . . . . . . . . . . . . . . . . . 461
12.2.1 MATLAB R© and Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
12.2.2 R and S-PLUS R© . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
12.3 High-Performance Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
12.4 Software for Statistical Applications . . . . . . . . . . . . . . . . . . . . . . . 472
12.5 Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
A Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
A.1 General Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
A.2 Computer Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
A.3 General Mathematical Functions and Operators . . . . . . . . . . . . . 482
A.4 Linear Spaces and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
A.5 Models and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
B Solutions and Hints for Selected Exercises . . . . . . . . . . . . . . . . . 493
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
132 3 Basic Properties of Matrices
It is easy to see that this measure has the consistency property (Exercise 3.27),
as a norm must. The Frobenius norm is sometimes called the Euclidean matrix
norm and denoted by ‖ · ‖E, although the L2 matrix norm is more directly
based on the Euclidean vector norm, as we mentioned above. We will usually
use the notation ‖ · ‖F to denote the Frobenius norm. Occasionally we use
‖ · ‖ without the subscript to denote the Frobenius norm, but usually the
symbol without the subscript indicates that any norm could be used in the
expression. The Frobenius norm is also often called the “usual norm”, which
emphasizes the fact that it is one of the most useful matrix norms. Other
names sometimes used to refer to the Frobenius norm are Hilbert-Schmidt
norm and Schur norm.
A useful property of the Frobenius norm that is obvious from the defini-
tion is
‖A‖F =
√
tr(ATA)
=
√
〈A,A〉;
that is,
• the Frobenius norm is the norm that arises from the matrix inner prod-
uct (see page 74).
From the commutativity of an inner product, we have ‖AT‖F = ‖A‖F. We
have seen that the L2 matrix norm also has this property.
Similar to defining the angle between two vectors in terms of the inner
product and the norm arising from the inner product, we define the angle
between two matrices A and B of the same size and shape as
angle(A,B) = cos−1
( 〈A,B〉
‖A‖F‖B‖F
)
. (3.233)
If Q is an n × m orthogonal matrix, then
‖Q‖F =
√
m (3.234)
(see equation (3.169)).
If A and B are orthogonally similar (see equation (3.191)), then
‖A‖F = ‖B‖F;
that is, the Frobenius norm is an orthogonally invariant norm. To see this, let
A = QTBQ, where Q is an orthogonal matrix. Then
‖A‖2F = tr(ATA)
= tr(QTBTQQTBQ)
= tr(BTBQQT)
= tr(BTB)
= ‖B‖2F.

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . ix
1. Linear Equations . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . 1
1.2 Gaussian Elimination and Matrices . . . . . . . . 3
1.3 Gauss–Jordan Method . . . . . . . . . . . . . . 15
1.4 Two-Point Boundary Value Problems . . . . . . . 18
1.5 Making Gaussian Elimination Work . . . . . . . . 21
1.6 Ill-Conditioned Systems . . . . . . . . . . . . . 33
2. Rectangular Systems and Echelon Forms . . . 41
2.1 Row Echelon Form and Rank . . . . . . . . . . . 41
2.2 Reduced Row Echelon Form . . . . . . . . . . . 47
2.3 Consistency of Linear Systems . . . . . . . . . . 53
2.4 Homogeneous Systems . . . . . . . . . . . . . . 57
2.5 Nonhomogeneous Systems . . . . . . . . . . . . 64
2.6 Electrical Circuits . . . . . . . . . . . . . . . . 73
3. Matrix Algebra . . . . . . . . . . . . . . 79
3.1 From Ancient China to Arthur Cayley . . . . . . . 79
3.2 Addition and Transposition . . . . . . . . . . . 81
3.3 Linearity . . . . . . . . . . . . . . . . . . . . 89
3.4 Why Do It This Way . . . . . . . . . . . . . . 93
3.5 Matrix Multiplication . . . . . . . . . . . . . . 95
3.6 Properties of Matrix Multiplication . . . . . . . 105
3.7 Matrix Inversion . . . . . . . . . . . . . . . 115
3.8 Inverses of Sums and Sensitivity . . . . . . . . 124
3.9 Elementary Matrices and Equivalence . . . . . . 131
3.10 The LU Factorization . . . . . . . . . . . . . 141
4. Vector Spaces . . . . . . . . . . . . . . . 159
4.1 Spaces and Subspaces . . . . . . . . . . . . . 159
4.2 Four Fundamental Subspaces . . . . . . . . . . 169
4.3 Linear Independence . . . . . . . . . . . . . 181
4.4 Basis and Dimension . . . . . . . . . . . . . 194
vi Contents
4.5 More about Rank . . . . . . . . . . . . . . . 210
4.6 Classical Least Squares . . . . . . . . . . . . 223
4.7 Linear Transformations . . . . . . . . . . . . 238
4.8 Change of Basis and Similarity . . . . . . . . . 251
4.9 Invariant Subspaces . . . . . . . . . . . . . . 259
5. Norms, Inner Products, and Orthogonality . . 269
5.1 Vector Norms . . . . . . . . . . . . . . . . 269
5.2 Matrix Norms . . . . . . . . . . . . . . . . 279
5.3 Inner-Product Spaces . . . . . . . . . . . . . 286
5.4 Orthogonal Vectors . . . . . . . . . . . . . . 294
5.5 Gram–Schmidt Procedure . . . . . . . . . . . 307
5.6 Unitary and Orthogonal Matrices . . . . . . . . 320
5.7 Orthogonal Reduction . . . . . . . . . . . . . 341
5.8 Discrete Fourier Transform . . . . . . . . . . . 356
5.9 Complementary Subspaces . . . . . . . . . . . 383
5.10 Range-Nullspace Decomposition . . . . . . . . 394
5.11 Orthogonal Decomposition . . . . . . . . . . . 403
5.12 Singular Value Decomposition . . . . . . . . . 411
5.13 Orthogonal Projection . . . . . . . . . . . . . 429
5.14 Why Least Squares? . . . . . . . . . . . . . . 446
5.15 Angles between Subspaces . . . . . . . . . . . 450
6. Determinants . . . . . . . . . . . . . . . 459
6.1 Determinants . . . . . . . . . . . . . . . . . 459
6.2 Additional Properties of Determinants . . . . . . 475
7. Eigenvalues and Eigenvectors . . . . . . . . 489
7.1 Elementary Properties of Eigensystems . . . . . 489
7.2 Diagonalization by Similarity Transformations . . 505
7.3 Functions of Diagonalizable Matrices . . . . . . 525
7.4 Systems of Differential Equations . . . . . . . . 541
7.5 Normal Matrices . . . . . . . . . . . . . . . 547
7.6 Positive Definite Matrices . . . . . . . . . . . 558
7.7 Nilpotent Matrices and Jordan Structure . . . . 574
7.8 Jordan Form . . . . . . . . . . . . . . . . . 587
7.9 Functions of Nondiagonalizable Matrices . . . . . 599
Contents vii
7.10 Difference Equations, Limits, and Summability . . 616
7.11 Minimum Polynomials and Krylov Methods . . . 642
8. Perron–Frobenius Theory . . . . . . . . . 661
8.1 Introduction . . . . . . . . . . . . . . . . . 661
8.2 Positive Matrices . . . . . . . . . . . . . . . 663
8.3 Nonnegative Matrices . . . . . . . . . . . . . 670
8.4 Stochastic Matrices and Markov Chains . . . . . 687
Index . . . . . . . . . . . . . . . . . . . . . . 705
5.2 Matrix Norms 279
5.2 MATRIX NORMS
Because Cm×n is a vector space of dimension mn, magnitudes of matrices
A ∈ Cm×n can be “measured” by employing any vector norm on Cmn. For
example, by stringing out the entries of A =
(
2 −1
−4 −2
)
into a four-component
vector, the euclidean norm on 4 can be applied to write
‖A‖ =
[
22 + (−1)2 + (−4)2 + (−2)2
]1/2
= 5.
This is one of the simplest notions of a matrix norm, and it is called the Frobenius
(p. 662) norm (older texts refer to it as the Hilbert–Schmidt norm or the Schur
norm). There are several useful ways to describe the Frobenius matrix norm.
Frobenius Matrix Norm
The Frobenius norm of A ∈ Cm×n is defined by the equations
‖A‖2F =
∑
i,j
|aij |2 =
∑
i
‖Ai∗‖22 =
∑
j
‖A∗j‖22 = trace (A∗A). (5.2.1)
The Frobenius matrix norm is fine for some problems, but it is not well suited
for all applications. So, similar to the situation for vector norms, alternatives need
to be explored. But before trying to develop different recipes for matrix norms, it
makes sense to first formulate a general definition of a matrix norm. The goal is
to start with the defining properties for a vector norm given in (5.1.9) on p. 275
and ask what, if anything, needs to be added to that list.
Matrix multiplication distinguishes matrix spaces from more general vector
spaces, but the three vector-norm properties (5.1.9) say nothing about products.
So, an extra property that relates ‖AB‖ to ‖A‖ and ‖B‖ is needed. The
Frobenius norm suggests the nature of this extra property. The CBS inequality
insures that ‖Ax‖22 =
∑
i |Ai∗x|2 ≤
∑
i ‖Ai∗‖
2
2 ‖x‖
2
2 = ‖A‖
2
F ‖x‖
2
2 . That is,
‖Ax‖2 ≤ ‖A‖F ‖x‖2 , (5.2.2)
and we express this by saying that the Frobenius matrix norm ‖
‖F and the
euclidean vector norm ‖
‖2 are compatible . The compatibility condition (5.2.2)
implies that for all conformable matrices A and B,
‖AB‖2F =
∑
j
‖[AB]∗j‖22 =
∑
j
‖AB∗j‖22 ≤
∑
j
‖A‖2F ‖B∗j‖
2
2
= ‖A‖2F
∑
j
‖B∗j‖22 = ‖A‖
2
F ‖B‖
2
F =⇒ ‖AB‖F ≤ ‖A‖F ‖B‖F .
This suggests that the submultiplicative property ‖AB‖ ≤ ‖A‖ ‖B‖ should be
added to (5.1.9) to define a general matrix norm.
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'I have taught courses using Meyer's text for two semesters now and I like the book even better than when I first read it. The text is just what I want for an
advanced level course in Linear Algebra for applied mathematicians and engineers. I plan to use it again.' William C. Brown, Michigan State University
'Meyer extensively treats traditional topics in matrix analysis and linear algebra. The text is well written, with the exact statements of important definitions and
theorems set off in gray boxes, surrounded by proofs, motivational discussions, many examples and historical notes, and 749 exercises. Meyer intentionally
leaves the 'scaffolding' in place to help the reader understand the development of the subject ... Included are a separate solutions manual and a CD-ROM
containing the entire text and solution manual in a searchable, hyperlinked (cross-referenced) pdf format. This CD-ROM, one of the few with this feature, makes
the package of even greater reference value.' J. D. Fehribach, CHOICE
'Carl Meyer's book is an outstanding addition to the vast literature in this area. Its most distinctive feature is a seamless integration of the theoretical,
computational, and applied aspects of the subject, which stems from the author's extensive experience in both teaching and research. The author's clear and
elegant expository style is enlivened by a generous sprinkling of historical notes and aptly chosen quotations from famous mathematicians, making this book a
delight to read. If this textbook will not succeed in awakening your students' interest in matrices and their uses, nothing else will.' Michele Benzi, Emory University
'I like how well thought out and organized this book is. I would recommend that anyone who teaches a course in linear algebra consider this text. Those who
choose not to adopt this text would still find it a handy reference a good reminder of some of the practical issues of linear algebra that working scientists must
consider. In my opinion, the stronger the students in the course, and the longer they are exposed to the text (it would be best in a two semester sequence) the
better they will appreciate this book and its spiral development of ideas.' Joel Foisy, MAA Online
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Nice book for advanced undergrad/grad students
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Textbook Binding: 700 pages
Publisher: SIAM: Society for Industrial and Applied Mathematics; Har/Cdr edition (February 15, 2001)
Language: English
ISBN-10: 0898714540
ISBN-13: 978-0898714548
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The fresh approach of this book introduces a variety of problems with clarity and informality. The focus on applications demonstrates how linear algebra can be
applied to real-life situations. Numerous examples, exercises and historical notes are provided, along with a CD-ROM containing a searchable copy of the
textbook and solutions.
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Matrix analysis and applied linear algebra: Carl D. Meyer: 9780898714548: Amazon.com: Books
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Comment One person found this helpful. Was this review helpful to you? Report abuse
The reason that I gave it 4 stars is because I feel like it sometimes to fails to live up to the
...
By Josiah on December 25, 2014
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Finding a good intermediate book on linear algebra
By matrixlearner on December 28, 2010
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A Model of Authorship
By Gus on November 9, 2006
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In my opinion it is better than Prof
This is a gem of a book! In my opinion it is better than
Prof. Strang's book. Its lucidly written. With illustrative
examples.
Published 1 year ago by Brato Chakrabarti
Could be better, but it is concise
and typically clear
Reasonable. Could be better, but it is concise and
typically clear.
Published on February 11, 2015 by Ryan D
Save your money
The entire book, including the solutions manual, is
available for free in PDF format (albeit with a
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Published on May 10, 2014 by Liam Cullen
Excellent!!!
Excellent book. The answers really help to know if
you're on right path. I will enjoy it for the rest of my life
Published on November 23, 2013 by Ernesto Ramírez A.
Good, clear, and well written
I'm an EE Control Systems student in my senior year.
I've already had basic classes in linear algebra and
have used LA for for many other courses.
Published on September 30, 2013 by Jimmerson
One of my favorite books on linear
alebgra
This must be one of my favorite books on linear
algebra. This is a beautifully produced book and
covers almost everything one might need for an
introductory to intermediate...
Published on March 31, 2013 by Herbert Sauro
perfect!
This is a good book for matrix analysis! Maybe the
best I ever read!The book is new and of good
quantity.
Published on October 19, 2012 by zz
a Masterpiece
Every book should be written like this. Absolutely
marvelous; a pleasure to read! It is theoretical, but
presented in an easy manner with many examples
that show the concepts...
Published on March 19, 2011 by Pushkin Kachroo
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Really nice text on applied linear algebra. There are many examples that involve physical phenomena and
interesting applications. However, it also provides the theoretical details necessary to really understand matrix
computations at a deeper level. Proofs are provided throughout and there are numerous problems (proofs and
applications) that go along with each topic. Even slightly more obscure or advanced topics are covered in
addition to the standard material. One nice feature is the historical information provided in footnotes throughout
the book, which make the motivation for certain topics more obvious.
One disappointing feature: operation counts for numerical computations are included throughout, however they
usually stand alone, with no algorithm given in the presentation. This is a little annoying since you would have
to come up with the algorithm on your own to understand why the operation counts are what they are.
However, since the focus is not on numerical computation, this still serves to motivate the use of one algorithm
over another in practical applications.
Overall, this is a detailed book with nice explanations. It could be used for lower-level undergraduate students,
but is probably better for advanced undergrads or grad students who haven't taken much linear algebra. It is
similar to Strang's book but deeper in coverage.
Yes No
This is a "solid" matrix theory book for engineers or scientists. The reason that I gave it 4 stars is because I feel
like it sometimes to fails to live up to the claim of being as casual it claims. For instance the idea of a projector
matrix onto a subspace of an operator is very abstractly described but then is not backed up with what I would
like to see in examples. So in my opinion, the book sometimes is too abstract without clear explanation.
However, overall, this is a very good text that will take you far in the application of linear algebra in a wide
variety of problems.
Yes No
This book provides many good chapters with excellent, clear, explanations and numerously well-constructed
examples. It can be used as a text for a basic introductory level student as well as a book serving those who
need to learn advanced topics in modern matrix/linear algebra. The student is introduced to the discipline with
the usual gaussian elimination techniques as well as basic subspaces but is quickly lead to linear
transformation, norms, inner products, orthogonal projections and eigenvaules and eigenvectors. As with many
books, the chapters lead to the study of Jordan Form. This is where this book excels over many others because
of its detailed explanations and examples in the preceding chapters.
My own experience of trying to understand the Jordan Form has lead me in search for numerous other books
on: invariant subspaces, complimentary subspaces, nilpotent matrices, range-null decomposition and projection
operators. The foundation on successfully understanding the Jordan Form is a good understanding of these
preceding subjects. Other books either skimp over the basic details of these abstract subjects (books that are
too advanced) or failed to link them in a way that culminate in a coherent proof of the Jordan Form. This book
handles this difficult proof in a clear logically manner, even with illustrations, because of the way these
preceding abstract subjects are introduced to the student.
Yes No
This book contains a comprehensive treatment on the topic of matrix analysis and applied linear algebra. The
concepts are clearly introduced and developed. It is rich with detailed proofs that are easy to follow. Results are
summarized and clearly grouped and marked for reference. As a researcher and a practitioner, I found this
book quite useful in explaining mathematical concepts without the need for a classroom instructor.
Besides, this book comes with a CD that contains a PDF version which makes it quite useful to port as a
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But the answers of exercises are perfect for studying
By Yuchen Wang on July 22, 2015
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Five Stars
By Amazon Customer on October 19, 2015
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Good depth and breadth
By engineering guy on September 26, 2010
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Five Stars
By Constantine K. on March 22, 2015
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reference. It is very rich with problem sets that add insight, both theoretically and practically. It is accompanied
by a solutions manual which strengthens comprehension.
I highly recommend this book. I think it deserves to be a model to follow for authorship in the digital age.
Yes No
I have not read it yet. But the answers of exercises are perfect for studying.
Yes No
Best of matrix analisys's book
Yes No
This is a good overall book that goes beyond your basic linear algebra texts such as Leon. If you're on the
fence about buying it, just google Carl Meyer. His website has a digital copy of the text, so you can check it out
and decide if it's worth buying. I commend the author for making the digital copies available, which is
uncommon for an author to do. So if you find the text useful, please buy it!
Oh, by the way, the textbook includes a cd with searchable pdf copies of the book and solutions manual. That
alone sets this text apart from most others.
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Very good condition.
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Matrix analysis and applied linear algebra: Carl D. Meyer: 9780898714548: Amazon.com: Books
https://www.amazon.com/Matrix-analysis-applied-linear-algebra/dp/0898714540/ref=sr_1_1?ie=UTF8&qid=1487179595&sr=8-1&keywords=matrix+analysis+and+applied+linear+algebra[2/22/2017 11:47:00 AM]
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