Summary of Matrix Analysis [@Horn1990].

## Chapter 2

P69 Lemma 2.1.8 Convergence of a sequence of unitary matrices

P86 Theorem 2.4.2 Cayley-Hamilton Theorem

P88 Corollary 2.4.4

P89 Theorem 2.4.6 Limit of the entries of a diagonalizable matrix

P94 Theorem 2.4.15 McCoy’s Theorem of simultaneously upper triangularization

P95 Problem 3 Generalization of Cayley-Hamilton Theorem by classical adjoint in matrices whose entries come from a commutative ring not a field usually defined for vector space

## Chapter 3

P132-134 3.2.2 Solving linear dynamic systems using Jordan Canonical Form

P135 Theorem 3.2.4.2 About commuting of nonderogatory matrices

P137 Section 3.2.5 Using Jordan form to analyze convergent matrices

Theorems(Corollaries) from page142-147 are all about relationships between characteristic polynomial and minimal polynomial. Below is a selected list some of them.

P142 Theorem 3.3.1 Existence of a unique minimum degree monic polynomial annihilating matrix A

P143 Corollary 3.3.4 Minimal polynomial divides characteristic polynomial

P145 Theorem 3.3.6 Representation of minimal polynomial

## Chapter 4

P176 Theorem 4.2.2 Rayleigh-Ritz Theorem

P179 Theorem 4.2.11 Courant-Fischer Theorem (Generalization of Rayleigh-Ritz)

Different variations of Interlacing Theorem are listed from page181 to 189. The point is that they are all applications of Courant-Fischer Theorem (except Theorem 4.3.10, which is not covered in the lecture).

## Chapter 7

P403 Corollary 7.2.4 Relationship between positive semidefinite and characteristic polynomial

P405 Theorem 7.2.6 Positive Semi-definite (PSD) matrix A can be written as the unique k-th power of another PSD B and they commute. (This is sort of a special case of nonderogatory matrices which is stated in p135. We may combining this part with section 3.2)

### Decomposition Theorem

P412 Theorem 7.3.2 Polar decomposition

P414 Theorem 7.3.4 A normal -> PU=UP

P415 Theorem 7.3.5 SVD

P417 Theorem 7.3.6 Existence of inexact SVD (Approaching a matrix from matrices not necessary in the same subset)

P420 Theorem 7.3.10 Generalization of Courant-Fischer Theorem

P431 Example 7.4.8 Approximating a matrix by rotation (Application of SVD and matrix form of least squares)

P435 Example 7.4.13 ‘Two-sided’ approximation by rotation (Another application of SVD and solving matrix form of least squares)

### Symmetric gauge function and its implications

P438 Theorem 7.4.24 Symmetric gauge function=unitarily invariant norm

P445 Theorem 7.4.45 Theorem of Kyfan

P447-450 A bunch of corollaries combined with majorization and interlacing theorem

### Schur product theorem and its implications

P459 Corollary 7.5.4 Fejer’s theorem on determining PSD matrices using schur product. (Recall that another way of determining PSD is by Characteristic polynomials P401.)

P459 Application 7.5.5-8 Solving elliptic PDE and uniqueness of solution.

P461 Corollary 7.5.9 PSD property for matrices with entry-wise polynomials.

## Chapter 8

Stochastic and doubly stochastic matrices

P527 Theorem 8.7.1 Birkhoff theorem: The set of all doubly stochastic matrices form a convex hull.

## Appendix B

P533 Convex geometry and convex functions. Set of established inequalities might be useful for analysis.