Page 10 - Matrices theory and applications

Contents

Preface vii

List of Symbols xiii

1Elementary Theory 1
1.1 Basics .. ... .. .. ... .. .. ... .. .. ... .. 1
1.2 Change of Basis .. .. ... .. .. ... .. .. ... .. 8
1.3 Exercises . ... .. .. ... .. .. ... .. .. ... .. 13

2 Square Matrices 15
2.1 Determinants and Minors .. .. .. ... .. .. ... .. 15
2.2 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Alternate Matrices and the Pfaffian ... .. .. ... .. 21
2.4 Eigenvalues and Eigenvectors . .. ... .. .. ... .. 23
2.5 The Characteristic Polynomial . .. ... .. .. ... .. 24
2.6 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Trigonalization . .. .. ... .. .. ... .. .. ... .. 29
2.8 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Exercises . ... .. .. ... .. .. ... .. .. ... .. 31

3 Matrices with Real or Complex Entries 40
3.1 Eigenvalues of Real- and Complex-Valued Matrices . . . 43
3.2 Spectral Decomposition of Normal Matrices .. ... .. 45
3.3 Normal and Symmetric Real-Valued Matrices . . . . . . 47