Corollary 1 2 9 If A is diagonalizable

Corollary 1. 2. 9 If A is diagonalizable and rank A=r, then A has at least one rxr nonsingular principal submatrix.

permutation matrix

Proof

Proof

Proof

Fact 1. 2. 10 p. 1 If A is mxn matrix and r is the size of the largest nonsingular submatrix. Then (i) rank A=r (ii) If B is a rxr nonsingular submatrix, then there are permutation matrices

Fact 1. 2. 10 p. 2 P, Q such that (iii) If , in addition, m=n and B is principal then may choose

Proof of Fact 1. 2. 10 p. 1

Proof of Fact 1. 2. 10 kth row p. 2

Proof of Fact 1. 2. 10 p. 3

Proof of Fact 1. 2. 10 p. 4

Proof of Fact 1. 2. 10 p. 5

Theorem 1. 2. 13 If m=n , and at least one of A or B is nonsigular, then AB and BA are similar

Proof of Theorem 1. 2. 13 p. 1

Proof of Theorem 1. 2. 13 p. 2

Corollary 1. 4. 3 If A is a real symmetric matrix of rank r then there is a permutation P and rxr nonsingular principal submatrix M s. t

Proof of Corollary 1. 4. 3 p. 1

Proof of Corollary 1. 4. 3 p. 2

Usual Inner Product of

Unitary U is said to be and equals i. e unitary if exists

Fact is unitary if and only if the columns of U form an orthonormal basis of proof: (see next page)

Real Orthogonal is real orthogonal if i. e A real orthogonal matrix is a real matrix which is unitary

Fact is real orthogonormal if and only if the columns of U form an orthonormal basis of proof: (see next page)

Fact A is unitarily diagonalized has a orthonormal basis consisting of eigenvectors of A proof: (in next page)

Fact A is diagonalizable has a basis consisting of eigenvectors of A proof: (in next page)

Theorem 1. 4. 1(The spectral Thm for Hermitian matrices) p. 1 ? (未證)

Theorem 1. 4. 1(The spectral Thm for Hermitian matrices) If A is real sysmmetric, then U can be chosen to be real orthogonal matrix p. 2