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## Über dieses Buch

In this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. With forty percent new material, this second edition is significantly different from the first edition. Newly added topics include: • Dunford decomposition, • tensor and exterior calculus, polynomial identities, • regularity of eigenvalues for complex matrices, • functional calculus and the Dunford–Taylor formula, • numerical range, • Weyl's and von Neumann’s inequalities, and • Jacobi method with random choice. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the École Normale Supérieure de Lyon.

## Inhaltsverzeichnis

### Chapter 1. Elementary Linear and Multilinear Algebra

This chapter is the only one where results are given either without proof, or with sketchy proofs. A beginner should have a close look at a textbook dedicated to linear algebra, not only reading statements and proofs, but also solving exercises in order to become familiar with all the relevant notions.

Denis Serre

### Chapter 2. What Are Matrices

In real life, a matrix is a rectangular array with prescribed numbers

n

of rows and

m

of columns (

n

×

m

matrix). To make this array as clear as possible, one encloses it between delimiters; we choose parentheses in this book. The position at the intersection of the

i

th row and

j

th column is labeled by the pair (

i

,

j

). If the name of the matrix is

M

(respectively,

A

,

X

, etc.), the entry at the (

i

,

j

)th position is usually denoted

m

i

j

(respectively,

a

i

j

,

x

i

j

). An entry can be anything provided it gives the reader information. Here is a the real-life example.

Denis Serre

### Chapter 3. Square Matrices

The essential ingredient for the study of square matrices is the determinant. For reasons given in Section 3.5, as well as in Chapter 9, it is useful to consider matrices with entries in a ring. This allows us to consider matrices with entries in(rational integers) as well as in

K

[

X

] (polynomials with coefficients in

K

).We assume that the ring of scalars

A

is a commutative (meaning that the multiplication is commutative) integral domain (meaning that it does not have divisors of zero:

ab

=0 implies either

a

= 0 or

b

= 0), with a unit denoted by 1, that is, an element satisfying 1

x

=

x

1 =

x

for every

x ∈≤

A

.

Denis Serre

### Chapter 4. Tensor and Exterior Products

Let

E

and

F

be

K

-vector spaces whose dimensions are finite. We construct their

tensor product

E

K

F

as follows.

Denis Serre

### Chapter 5. Matrices with Real or Complex Entries

A matrix

$$M \in {M_{n \times m}}(k)$$

is an element of a vector space of finite dimension

n

2. When

K

=

$${\mathbb{R}}$$

or

K

=

$${\mathbb{C}}$$

, this space has a natural topology, that of

K

nm

. Therefore we may manipulate such notions as open and closed sets, and continuous and differentiable functions.

Denis Serre

### Chapter 6. Hermitian Matrices

We recall that

$${\left\|\cdot\right\|_2}$$

denotes the usual Hermitian norm on :

$$\mathbb {C}$$

$${\left\| x \right\|_2}: = {\left( {\sum\limits_{j = 1}^n {{{\left| {{x_j}} \right|}^2}} } \right)^2}$$

Denis Serre

### Chapter 7. Norms

In this chapter, the field

K

is always

$$\mathbb {R}$$

or

$$\mathbb {C}$$

and

E

denotes

K

n

. The scalar (if

K

=

$$\mathbb {R}$$

) or Hermitian (if

K

=

$$\mathbb {C}$$

) product on

E

is denoted by

$$\left\langle {x,y} \right\rangle : = {\Sigma _j}{\bar x_j}{y_j}.$$

Denis Serre

### Chapter 8. Nonnegative Matrices

In this chapter matrices have real entries in general. In a few specified cases, entries might be complex.

Denis Serre

### Chapter 9. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction

In this chapter we consider only

commutative integral domains A

(see Chapter 3). Such a ring

A

can be embedded in its field of fractions, which is the quotient of

$$A \times (A\backslash \left\{ 0 \right\})$$

by the equivalence relation

$${\text{(a,b)}} {\mathcal{R}}{\rm {(c,d)}} \Leftrightarrow {\rm ad = bc}.$$

The embedding is the map

$$\begin{array}{l}\\a \mapsto (a,1) \\\end{array}$$

.

Denis Serre

### Chapter 10. Exponential of a Matrix, Polar Decomposition, and Classical Groups

Polar decomposition and exponentiation are fundamental tools in the theory of finite-dimensional Lie groups and Lie algebras. We do not consider these notions here in their full generality, but restrict attention to their matricial aspects.

Denis Serre

### Chapter 11. Matrix Factorizations and Their Applications

The techniques described below are often called

direct solving methods.

Denis Serre

### Chapter 12. Iterative Methods for Linear Systems

In this chapter the field of scalars is

K

= ℝ or ℂ.

Denis Serre

### Chapter 13. Approximation of Eigenvalues

The computation of the eigenvalues of a square matrix is a problem of considerable difficulty. The naive idea, according to which it is enough to compute the characteristic polynomial and then find its roots, turns out to be hopeless because of Abel’s theorem, which states that the general equation

P

(

x

) = 0, where

P

is a polynomial of degree d ≥ 5, is not solvable using algebraic operations and roots of any order. For this reason, there exists no direct method, even an expensive one, for the computation of Sp(

M

).

Denis Serre

### Backmatter

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