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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 1539

Book Review

Matrix Mathematics: Theory, Facts, and Formulas with

Application to Linear Systems TheoryDennis S. Bernstein

(Princeton University Press, 2005). Reviewed by Michael K. Sain

It was 0 20 F in South Bend, IN, when I began my appointment as

a faculty member at the Department of Electrical Engineering, Univer-

sity of Notre Dame, Notre Dame, IN, early in January 1965. Although

I was primed for the cold weather, my 1958 black Dodge hardtop was

not. We did not have a garage, nor even a snow shovel. So I usually

called Fred in the morning for a jump start. Later, we decided to sell

the car to Fred. At least, he would be able to get it going.

But that was only part of my startup problems. I was a freshly

minted Assistant Professor, and the students thought I knew, or should

know, everything. Yet a month prior, I had only been a graduate stu-

dent at the University of Illinois, Urbana; and so far I had learned just a

small part of everything. One had ones class notes, a couple of years

hard copies of the IEEE TRANSACTIONS ONAUTOMATICCONTROL, the

textbooks from my courses, and a small but treasured collection of ref-erence books. From these sources, plus the University of Notre Dame

library, I was setting out to interpret the world of knowledge and re-

search for my students. It was certainly exciting, but also somewhat

daunting.

Linear systems were king, and the Information Dissemination

Committee, storied predecessor of the TRANSACTIONSEditorial Board,

was in full swing. Kalman was leading a strong surge of ideas such

as reachability, controllability, observability, Lyapunov stability, and

linear-quadratic-Gaussian optimal control theory, together with its

dual filter and the separation principle. Discrete- and continuous-time

cases were studied in parallel or in series, according to the wonts of

the author and/or the professor, and both viewpoints held sway.

However, one thing was very definitely true: We had a great needto know a great deal about a great number of matrices. The penetra-

tion of state variablesalive and well after hundreds of years in

theoretical and applied mechanicsinto the control systems mindset

meant that matrices were becoming asine qua nonin the toolkit of the

young systems engineer. Whether you were using and , the

notation of Kalman, or the more generic and , you were using

matrices both in your theoretical investigations and in your computa-

tions. Onespoke of the scalar caseor the matrix case, andclassified

research results according to whether they were single-inputsingle-

output (SISO) or multiple-inputmultiple-output (MIMO); a great deal

of interest attached to whether or not

or

depended on the indepen-

dent index, be it time or a step counter.

Whatever. Matrices were everywhere. One of my favorite memories

is that of Prof. Wonham talking animatedly about flying matricesand steaming matrices. It would have been a mark of distinction

to own an encyclopedic reference on matrices in those days. However,

as luck would have it, we could not.

Those of us from that generation, and maybe a fair number from

the succeeding generation, would assemble a collection of books and

papers on the subject, hoping to get a finite covering of the needed

materialat least that portion which related closely to our research and

teaching efforts. Perhaps that might have been a dozen or 20 items, if

The reviewer is with the Department of Electrical Engineering, University ofNotre Dame, Notre Dame, IN 46556 USA.

Digital Object Identifier 10.1109/TAC.2007.902261

one includes both favoritebooks andpreferred papers, occupying a foot

or so on the bookshelf and maybe half of a file drawer.

But it most certainly would not have been 820 items, which is the

number of entries in the bibliography of Bernsteins Matrix Math-ematics (Princeton University Press: Princeton, NJ, 2005). That is

about 50 times the number of my favorite references, whose collection

began some 50 years before the publication of this new book. Subtitled

Theory, Facts, and Formulas with Application to Linear Systems

Theory, Matrix Mathematics is the encyclopedic work on matrices

that we all sought 50 years ago. It did not take the entire five decades

to bring into being, but consumed nonetheless a good portion of it15

years of detailed effort. One is reminded of Websters first dictionary,

and the professors joke that Noah would not have received tenure,

because he had only worked on one publication.

Occupying 48 of the book pages, the bibliography contains an

estimated 1400 names. No doubt many of these names will be familiar

to researchers and teachers who were active in this half-century:

Aitken, Altmann, Anderson, Aplevich, Barnett, Bellman, Brewer,

Brockett, Desoer, Francis, Fuhrmann, Greub, Hahn, Halmos, Hautus,

Householder, Jury, Kucera, Kwakernaak, Laub, Leake, Luenberger,

MacDuffee, Macfarlane, Newcomb, Popov, Siljak, Wolovich, and

Wonham, to name just a few. Each of us would take note of a different

subset of the overall list, of course. But most would finish a careful pe-

rusal with an in-depth impression that there are hundreds more names

that we do not knowpersons that we never had the opportunity to

meet.

The bibliography is not the only helpful reference structure in the

book. A lengthy and detailed index of more than 100 pages is also pro-

vided. Especially notable is the five-page index of symbols which ap-

pears at the beginning of the index. Together with the 15-page section

on special symbols in the front of the book, which section is comple-mented by a four-page treatment on conventions, notations, and ter-

minology, these features go a long way toward helping the reader to

decipher mathematical expressions which he or she may encounter on

a trek through the inner reaches of the work. Overall, out of approx-

imately 750 pages, some 190 are devoted to such matters of bibliog-

raphy, indexing, and notation. That is about one-quarter of the book.

Thereader,therefore, hasan excellentchance to determine if the idea

which is sought is to be found in the volume; and, of course, that is one

way in which a person may wish to make use ofMatrix Mathematics.

Other ways are possible. Bernstein suggests that the book is intended

to be useful to at least four groups of readers. Now that linear algebra

is a standard course in the mathematical sciences and engineering, he

points out that students in such venues can use the book to expand the

scope of their coursetextbook. However, only a portion of the results in

Matrix Mathematicsappear together with a proof. In this sense, there

will be a difference in readability relative to the context in a textbook.

Theremaining results, without proofs, are of two types: those with cita-

tions to one of the 820 references and those with no such citation. Both

of these types would have the character of homework problemsthe

former requiring some digging into the literature and the latter being

littleresearch problems. This is the second way that the author suggests

that the book be used: to supply exercises to augment standard mate-

rial in matrix courses. Teachers considering such usage will wish to

make sure that their students have library or other internet access to the

references cited. Most large libraries will be able to provide such ac-

cess; some smaller libraries may be less convenient in this regard. The

third possible usage offered by the author is for general reference for

0018-9286/$25.00 2007 IEEE

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1540 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007

researchers in mathematics, statistics, physics, and engineering. This,

of course, is the way we would have wanted to use it in the 1960s. Fi-

nally, there is the use of the book as a reference in the applications.

Here, perhaps, the absence of proofs is less of a difficulty, although the

occasional absence of citations may inhibit further exploration around

a result of particular interest.

But one of the most interesting ways to useBernsteinsMatrix Math-

ematicsis toperuseit. If you fancy yourself as being well informedabout matrices and their use in systems theory and applications, you

may wish to page through the book to determine whether you already

know everything that is in it. Not many, if any, will be able to pass that

test. Also very enjoyable is seeing whether one canprovethe results

which are not familiar quickly, or whether one will getbogged down

and have to do some work to verify the particular property. Then again,

if you happen to be someone who is an expert in some aspect of matrix

theory or application, you may wish to see how well the book treats that

aspect. In this type of exploration, one oftenfinds that the book is in-

deed like an encyclopedia, wherein onefinds the basics of a given idea,

and maybe even a few of thefine details, but certainly not an exhaus-

tive treatment. Amazingly, this reminds us that even an encyclopedic

treatment of matrices puts only a dent in the even more encyclopedicsubject of matrices.

In perusingMatrix Mathematics, the reader will encounter chapters

on basic matrix properties, matrix classes and transformations, matrix

polynomials and rational transfer functions, matrix decompositions,

generalized inverses, Kronecker and Schur algebra, positivesemidef-

inite matrices, norms, functions of matrices and their derivatives, the

matrix exponential and stability theory, and linear systems and control

theory.

In each chapter, one may encounter theorems, corollaries, lemmas,

propositions, and facts. By far, the most common encounter is with

facts. For example, in Chapter 2, on basic matrix properties, Fact

2.14.12 states that, if and are real or complex matrices with

rows and

columns, and if

is nonsingular, then

0

0

(1)

This is an example of a result stated without proof and without citation

to one of the bibliographical entries. Thus, we leave its veri fication to

the reader as an exercise.

Although Matrix Mathematics was not available in the 1960s, it is

of some historical interest to make a few comparisons with a work

which was. One solid beginning that a young faculty member could

have made was to purchase the two-volume work of Gantmacher on

The Theory of Matrices[1]. Books were cheaper then, of course; the

price on my shelf copy is marked $6.00 per volume. It is revealing to

browse through Gantmachers contents, in the light of what has tran-

spired in the intervening 50 years. One finds, for example, material oncomputational questions, including the standard Gaussian elimination

and the Fadeeva algorithm, which may be contrasted with the material

on singular value decompositions, in thefifth chapter of the Bernstein

book. By now, most everyone knows how that transition took place.

Polynomial matrices receive considerable attention, in chapters four

and six of Gantmachers firstvolume, and in the fourth chapter of Bern-

stein. With hindsight one can foresee the matrix fraction representation

0

and the system matrix formulation of Rosen-

brock, which led to generalizations of the classical ideas of poles and

zeros of a transfer function to poles and zeros of a matrix transfer func-tion and to poles and zeros of the underlying state dynamical system. In

this regard, one of the curious omissions ofMatrix Mathematicsis that

there is no item in the bibliography from among Rosenbrocks many

works. To be sure, in Section 12.10, on system zeros, the author does

introduce theRosenbrock system matrix,but without a citation. This

is likelyan accidental oversight, as many of us still carefully watchover

our copy of State Space and Multivariable Theory [2]. Another com-

parison can be made between the fifth chapter of Gantmachers first

volume, on functions of matrices, and thetenth chapter ofMatrix Math-

ematics: Functions of Matrices and Their Derivatives, together with the

treatment of the matrix exponential in Chapter 11. Gantmachers text

clearly presaged ideas which have now become commonplace.

In any encyclopedic treatment, the reader will find topics which heor she wishes had received greater treatment. This is unavoidable, and

is of the very nature of such an approach. For instance, the compu-

tational aspects of matrix operations are crucial to the successful ap-

plication of the idea of a matrix. The author comments upon this fact

in the third paragraph of the Preface, and even gives 17 related refer-

ences to the Bibliography. Yet the volume does not make any detailed

attempt to address issues of computation. Along the same lines, the

reviewer approached the seventh chapter, on Kronecker and Schur al-

gebra, with great interest, because of a long-standing theoretical and

applied experience with the notion of tensors. Indeed, the notion of

multilinear mappings, together with their universal expression in the

tensor algebra, gives rise to matrices in nonlinearcontexts; and this

can extend the impact of a book on matrices. Although the Kronecker

and Schur products are important special cases of the tensor product,theydo not really convey the underlying structure of the ideain the way

that one could have hoped. To be sure, each reader could likely report

some such similar reaction to this extensive work.

Nonetheless, BernsteinsMatrix Mathematicsis a monumental con-

tribution which takes a serious step towardsystematizing an abso-

lutely huge area of theory and application. The reviewer wishes to ap-

plaud this effort, and to invite the reader to perusethe work for him-

self or herself. It is fun to match wits with this compilation, and in the

process to expand ones horizon even further. Enjoy!

REFERENCES

[1] F. R. Gantmacher, The Theory of Matrices. New York: Chelsea,1959, vol. I and II.[2] H. H. Rosenbrock, State Space and Multivariable Theory. New York:

Wiley, 1970.

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