Jordan Canonical Form: Theory and Practice (Synthesis Lectures on Mathematics and Statistics, 6) by Steven Weintraub (PDF)

Description

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V – > V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1. We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J, and a refinement, the labeled eigenstructure picture (lESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader. Table of Contents: Fundamentals on Vector Spaces and Linear Transformations / The Structure of a Linear Transformation / An Algorithm for Jordan Canonical Form and Jordan Basis

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐an incredible (and understandable) presentation of an extremely abstruse mathematical conception. the authors did a wonderful job not only with theory, but with their multiple examples. i have only two complaints: there ARE a number of typo’s/errors (which the author should edit with an errata list) and why not provide the answers to the even numbered problems as well? you can check your answer anyway by seeing if AP=PJ, so why not save th reader the time and trouble. Overall a great book! I have ordered the author’s other book on Jordan matrices because I liked this one so muchj

⭐Don’t miss this thin book on the the Jordan Canonical Form (JCF). But, beware. It is full of typos and plagued with an army of bizarre acronyms (lESP, BDBUTCD…) and heavy abbreviations, as “alg mult(lambda)” instead of “multiplicity of lambda”. It has no bibliography, nor does it mention other methods to get the JCF. But, anyway, as it is, this monograph is an useful tool, in a world crowded with too many books on Linear Algebra, most dealing with JCF in a hurry. It is written in a simple and down to earth style. The JCF J(A) of a complex square matrix A always exists, it is either diagonal, or the sum of a diagonal matrix D (with the eigenvalues of A as the elements D_(i,i)) and a nilpotent matrix N, (with every component null, excepting N_(i,i+1) = 1 or 0). The link between A and J(A) is a (non unique) regular matrix P such that AP=PJ(A). There is a well-known method to obtain J(A), (see Smale-Hirsch’s book

⭐). It is well-known (but not mentioned) that the JCF of a real matrix A is needed to express the general solution of the differential equation x’=Ax, which is x(t)=Real[Pexp(tJ(A))w], (where w is an arbitray real vector and where Real[u+iv]=u, for a complex vector z=u+iv). The exponential exp(tJ(A)) is not hard to compute; the true difficulty is to find the matrix P. In the general case, that’s equivalent to obtain a good basis in the complex vector space V, where A acts on; P is then the change of bases matrix. To get such a holy basis, we must express V as a sum af cyclic sub-spaces and then collect the so called cyclic basis on them (each “Jordan cell” in J(A) corresponds to such a cyclic basis). Old books I once studied, (like Lang’s

⭐, Birkhoff-MacLane’s

⭐, or Herstein’s

⭐), do not give a clear algorithmic procedure to obtain such a basis on V. Happily, this work does exactly that. However, it skips explaining the JCF when A is nilpotent; nor does it speak of cyclic bases and cyclic sub-spaces, taking care only of the algorithm and missing the opportunity to give a complete picture of the subject. In the positive side: there are many worked examples and 60 exercises (half of them solved). An updated, but shorter exposition of the algorithm, by the same author, is Weintraub’s

⭐. While there are less typos, no further improvemement have I noticed, excepting two additional worked examples.Other books where the method to obtain P is quite well explained are: Hoffman-Kunze’s

⭐; S. Lipschutz’s

⭐; Hefferon’s

⭐and Pontrjagin’s

⭐.

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