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 rank A=r 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 p.2 kth row

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 unitary if exists and equals i.e

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) p.2 If A is real sysmmetric, then U can be chosen to be real orthogonal matrix