id: 06492266
dt: b
an: 2016c.00797
au: Liesen, Jörg; Mehrmann, Volker
ti: Linear algebra.
so: Springer Undergraduate Mathematics Series. Cham: Springer (ISBN
978-3-319-24344-3/pbk; 978-3-319-24346-7/ebook). xi, 324~p. (2015).
py: 2015
pu: Cham: Springer
la: EN
cc: H65
ut: textbook; Gaussian elimination; determinants; eigenvalues; eigenvectors;
Cayley-Hamilton theorem; vector space; linear transformation; bilinear
form; Euclidean space; unitary spaces; adjoint endomorphisms; Schur
triangularisation; Jordan canonical form; functions of matrices;
differential equations; Sylvester’s law of inertia; singular value
decomposition; image compression; linear matrix equations; Kronecker
product; Google’s PageRank algorithm; Markov chain; least squares;
numerical examples
ci: Zbl 1305.15002
li: doi:10.1007/978-3-319-24346-7
ab: This book is a translation of the second edition of the authors’ German
version [Heidelberg: Springer Spektrum (2015; Zbl 1305.15002)]. It
provides a good introductory undergraduate course at an intermediate
level which the authors describe as: “matrix-oriented, \dots
presenting a rather complete theory (including all details and proofs)
while keeping an eye on the applicability of the results". The first
third of the book restricts itself to matrices and covers standard
topics such as: solution of linear equations, Gaussian elimination and
echelon form; determinants; eigenvalues, eigenvectors and the
Cayley-Hamilton theorem. This is done is a concrete and classical way.
From this point on, the material becomes more abstract: abstract vector
spaces and linear transformations; bilinear forms; Euclidean and
unitary spaces; adjoint endomorphisms; Schur triangularization; the
Jordan canonical form. The last four chapters of the book deal with
functions of matrices and application to differential equations;
diagonalization of normal endomorphisms and geometry of Sylvester’s
law of inertia; the singular value decomposition and image compression;
and solution of linear matrix equations using the Kronecker product.
The translation reads well and the exposition is clear. From the
beginning, the authors motivate the material with interesting examples
such as Google’s PageRank algorithm (eigenvectors), car insurance
(Markov chains) and prediction via least squares, and although the book
emphasizes theory rather than computation, the theoretical results are
liberally illustrated with simple numerical examples. Computational
issues are not directly addressed, but the text includes short
“MATLAB-Minutes" which are exercises providing an informal
introduction to the use of MATLAB in linear algebra, including hints
about how care may be needed when working in finite precision.
rv: John D. Dixon (Ottawa)