ALL CHAPTERS COVERED
,Contents
Preface v
Notation ix
1 Basic Operations I
2 Linear Equations 47
3 Kronecker Product 71
4 Traces, Determinants and Hyperdeterminants 99
5 Eigenvalues and Eigenvectors 142
6 Spectral Theorem 205
7 Commutators and Anticommutators 217
8 Decomposition of Matrices 241
9 Functions of Matrices 260
10 Cayley-Hamilton Theorem 299
11 Hadamard Product 309
12 Norms and Scalar Products 318
13 vec Operator 340
14 Nonnormal Matrices 355
15 Binary Matrices 365
16 Star Product 371
17 Unitary Matrices 377
18 Groups, Lie Groups and Matrices 398
vii
,viii Contents
19 Lie Algebras and Matrices 439
20 Braid Group 466
21 Graphs and Matrices 486
22 Hilbert Spaces and M utually Unbiased Bases 496
23 Linear Differential Equations 507
24 Differentiation and Matrices 520
25 Integration and Matrices 535
Bibliography 547
Index 551
, Notation
is defined as
belongs to (a set)
i does not belong to (a set)
n intersection of sets
u union of sets
0 empty set
TcS subset T of set S
SnT the intersection of the sets S and T
SuT the union of the sets S and T
HS) image of set S under mapping /
f 09 composition of two mappings ( / o g)(x) = f(g (x ))
set of natural numbers
set of natural numbers including 0
set of integers
set of rational numbers
set of real numbers
K+ set of nonnegative real numbers
C set of complex numbers
Kn n-dimensional Euclidean space
space of column vectors with n real components
Cn n-dimensional complex linear space
space of column vectors with n complex components
H Hilbert space
Sn symmetric group on a set of n symbols
»(* ) real part of the complex number z
9 (* ) imaginary part of the complex number z
\z\ modulus of complex number z
\x + iy\ = (x2 + ?/2) 1/2, x , y € R
X column vector in Cn
XT transpose of x (row vector)
0 zero (column) vector
norm
x •y = x*y scalar product (inner product) in C n
x x y vector product in E 3
IX
,Contents
Preface v
Notation ix
1 Basic Operations I
2 Linear Equations 47
3 Kronecker Product 71
4 Traces, Determinants and Hyperdeterminants 99
5 Eigenvalues and Eigenvectors 142
6 Spectral Theorem 205
7 Commutators and Anticommutators 217
8 Decomposition of Matrices 241
9 Functions of Matrices 260
10 Cayley-Hamilton Theorem 299
11 Hadamard Product 309
12 Norms and Scalar Products 318
13 vec Operator 340
14 Nonnormal Matrices 355
15 Binary Matrices 365
16 Star Product 371
17 Unitary Matrices 377
18 Groups, Lie Groups and Matrices 398
vii
,viii Contents
19 Lie Algebras and Matrices 439
20 Braid Group 466
21 Graphs and Matrices 486
22 Hilbert Spaces and M utually Unbiased Bases 496
23 Linear Differential Equations 507
24 Differentiation and Matrices 520
25 Integration and Matrices 535
Bibliography 547
Index 551
, Notation
is defined as
belongs to (a set)
i does not belong to (a set)
n intersection of sets
u union of sets
0 empty set
TcS subset T of set S
SnT the intersection of the sets S and T
SuT the union of the sets S and T
HS) image of set S under mapping /
f 09 composition of two mappings ( / o g)(x) = f(g (x ))
set of natural numbers
set of natural numbers including 0
set of integers
set of rational numbers
set of real numbers
K+ set of nonnegative real numbers
C set of complex numbers
Kn n-dimensional Euclidean space
space of column vectors with n real components
Cn n-dimensional complex linear space
space of column vectors with n complex components
H Hilbert space
Sn symmetric group on a set of n symbols
»(* ) real part of the complex number z
9 (* ) imaginary part of the complex number z
\z\ modulus of complex number z
\x + iy\ = (x2 + ?/2) 1/2, x , y € R
X column vector in Cn
XT transpose of x (row vector)
0 zero (column) vector
norm
x •y = x*y scalar product (inner product) in C n
x x y vector product in E 3
IX
