Ebook: Similarity of matrices over a ring

I n t r o d u c t i o n ................. 1
Chapter 1: The Module Associated to a Matrix ........... 11
1. Definition of the Module ......................... 11
2. The Annihilator of M in R [ A ] ................ 12
3. Similarity of Matrices and Isomorphism
of Modules .......................................... 17
4. Properties of the Rings A and A ......... 19
Chapter 2: Lattices Over Orders ............... 25
1. Definitions and Basic Results .................. 25
2. The Jordan-Zassenhaus Condition ..... ........... 27
3. Genera of Lattices ................................. 33
Chapter 3: The Latimer-MacDuffee Correspondence . . . 40
1. The Classical Method .............................. 40
2. The Relationship of Modules and Ideals. . . . 47
Chapter 4: Block Triangular Form. ....................... 53
1. Transforming a Matrix into Block Triangular
Form ................................................. 53
2. Similarity of Matrices in Block Triangular
Form .................................................. 63
Chapter 5: Determination of Similarity .................. 69
1. Introductory R e m a r k s ........................... . 69
2. Form 1: £(X) has distinct irreducible
factors .................................................. 69
3. Form 2: All the roots of f(A) are the
same, f(A) = (A - a)n .............................. 71
4. Form 3: f(A) = g(A)(A - a) , and g(a) ^ 0 . 81
5. Form 4: f(A) = g(A)h(A) and h(A) has
degree 2 ............................................... 85
6 . Summary .................................................. 93
Bibliography ....................................................... 98
V i t a ................................................................ 101