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~。