英文版矩阵分析考试要点

inner product：

vector norm：

Matrix norm： operator norm

The l 2 norm is a matrix norm (Frobenius norm)

Proof We just verify the submultiplicative. Using Cauchy-Schwarz inequality, we have:

The l 1 norm is a matrix norm.Proof We just verify the submultiplicative.

Thel ∞ norm is not a matrix norm.Proof Consider the matrix:

The maximum column sum norm ||| · |||1 is deduced by the l1 norm. Proof ：

The maximum row sum norm ||| · |||∞ is deduced by the l ∞ norm. Proof ：

The spectral norm ||| · |||2 is deduced by the l 2 norm.

Proof

∑

=i

ij j

a A ||max ||||||1∑=∞j

ij i

a A |

|max ||||||{}

A

A of eigenvalue an is A *=λλ:max ||||||2

The matrix is called diagonalizable if A is similar to a diagonal matrix.

A matrix is diagonalizable iff A has n linearly independent eigenvectors.

If U is unitary, compute and |||U|||2 :solution

n

M A ∈n

M A

∈)

(U ρ1)}(|:max{|)(=∈=U U σλλρ{}

1

)(:max ||||||*2=∈=U U U σλλ

Minimal Polynomials:Let A ∈ Mn. Then there exists a unique monic annihilate polynomial q A (x) of minimum degree. If p(x) is any annihilate polynomial, then q A (x) divides p(x). [remarks: if p(A)=0, then p(x) is called an annihilate polynomial of A.“monic ”means the highest order coefficient of a polynomial is ‘1’]The polynomial q A (x) is called the minimal polynomial The Jordan canonical Form:Let λ∈C. A Jordan block J k (λ) is a k × k upper triangular matrix of the form

Every matrix is similar to a unique Jordan canonical form .

, So

i

k A

A A A A ⊕=⎥⎥⎥⎥

⎦

⎤⎢⎢⎢

⎢⎣

⎡

2

1~⎥⎥

⎥⎦

⎤⎢⎢⎢⎣⎡=i i

i i T A λλ *~i k T T T T A ⊕=⎥

⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣

⎡ 21~