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Front matter and contents of a lecture-notes text on linear algebra by G. Donald Allen of Texas A&M University, dated September 22, 2003. It lists ten chapters: vector spaces, matrices and linear systems, eigenvalues, unitary matrices, Hermitian and normal matrices, factorizations (PLU, QR, least squares), Jordan normal form, variational eigenvalue characterizations, and nonnegative and stochastic matrices. It is a third-party text kept in Phil's linear algebra folder.
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Lectures on Linear Algebra and Matrices
G. Donald Allen
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
September 22, 2003
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Contents
1 Vectors and Vector Spaces 7
1 . 1 V e c t o r S p a c e s........................... 7
1.1.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Linear independence and linear dependence . . . . . . . . . . 131 . 3 B a s e s ............................... 1 5
1 . 4 E x t e n s i o n t o a b a s i s ....................... 1 6
1 . 5 D i m e n s i o n............................. 1 71 . 6 N o r m s............................... 2 31 . 7 O r d e r e d B a s e s .......................... 3 01 . 8 E x e r c i s e s .............................. 3 2
2 Matrices and Linear Algebra 35
2 . 1 B a s i c s ............................... 3 5
2 . 2 L i n e a r S y s t e m s .......................... 4 42 . 3 R a n k ................................ 5 12.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.1 An important equality for matrix multiplication and
t h e i n n e r p r o d u c t ..................... 6 3
2 . 4 . 2 T h e L e g e n d r e P o l y n o m i a l s ................ 6 42 . 4 . 3 O r t h o g o n a l m a t r i c e s................... 6 6
2 . 5 D e t e r m i n a n t s ........................... 6 7
2 . 5 . 1 M i n o r s a n d D e t e r m i n a n t s ................ 7 2
2 . 6 P a r t i t i o n e d M a t r i c e s ....................... 7 82 . 7 L i n e a r T r a n s f o r m a t i o n s ..................... 8 02 . 8 C h a n g e o f B a s i s .......................... 8 32 . 9 A p p e n d i x A – S o l v i n g l i n e a r s y s t e m s ............. 8 52 . 1 0 E x e r c i s e s ............................. 8 8
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4 CONTENTS
3 Eigenvalues and Eigenvectors 95
3 . 1 M a t r i x N o r m s ........................... 9 53.2 Convergence and perturbation theory . . . . . . . . . . . . . 993 . 3 E i g e n v e c t o r s a n d E i g e n v a l u e s ..................1 0 33 . 4 T h e H a m i l t o n - C a y l e y T h e o r e m .................1 1 73 . 5 S i m i l a r i t y .............................1 2 33.6 Equivalent norms and convergent matrices . . . . . . . . . . . 1313 . 7 E x e r c i s e s .............................1 3 53 . 8 A p p e n d i x A............................1 3 83 . 9 A p p e n d i x B............................1 4 0
3.9.1 In fin i t e S e r i e s.......................1 4 0
4U n i t a r y M a t r i c e s 1 4 7
4 . 1 B a s i c s ...............................1 4 74 . 2 S c h u r ’ s t h e o r e m ..........................1 5 34 . 3 E x e r c i s e s .............................1 5 8
5 Hermitian Theory 161
5.1 Diagonalizability of Hermitian Matrices . . . . . . . . . . . . 1615 . 2 F i n d i n g e i g e n v e c t o r s . ......................1 6 65.3 Positive de fin i t e m a t r i c e s ....................1 7 0
5 . 4 E x e r c i s e s .............................1 7 0
6 Normal Matrices 171
7 Factorization Theorems 175
7 . 1 T h e P L U D e c o m p o s i t i o n .....................1 7 5
7.2LRLRLRf a c t o r i z a t i o n .........................1 8 3
7.3 The QRa l g o r i t h m ........................1 8 5
7 . 4 L e a s t S q u a r e s...........................1 9 1
8 Jordan Normal Form 195
8 . 1 M i n i m a l P o l y n o m i a l s.......................1 9 5
8 . 2 I n v a r i a n t s u b s p a c e s........................1 9 8
8 . 3 T h e J o r d a n N o r m a l F o r m ....................2 0 48 . 4 C o n v e r g e n t m a t r i c e s .......................2 0 88 . 5 E x e r c i s e s .............................2 0 9
CONTENTS 5
9 Hermitian and Symmetric Matrices 211
9.1 Variational Characterizations of Eigenvalues . . . . . . . . . . 2159 . 2 M a t r i x i n e q u a l i t i e s........................2 1 7
9 . 3 E x e r c i s e s .............................2 1 8
10 Nonnegative Matrices 219
10.1 De fin i t i o n s ............................2 1 9
1 0 . 2G e n e r a l T h e o r y..........................2 2 0
1 0 . 3M e a n E r g o d i c T h e o r e m .....................2 2 41 0 . 4I r r e d u c i b l e M a t r i c e s .......................2 2 7
10.4.1 Sharper estimates of the maximal eigenvalue . . . . . 233
1 0 . 5S t o c h a s t i c M a t r i c e s ........................2 3 6
6 CONTENTS