Theorems norms

Question 1

Let A ∈ C^n×n, for any consistent matrix norm || · ||

Answer 1

ρ(A) ≤ ||A||

Question 2

||A||1=

Answer 2

max ||Ax||1
x=!0 ||x||1

= max (1-j-n) SUM^m_i=1: |aij|
=max col sum

Question 3

||A||∞=

Answer 3

max ||Ax||∞
x=!0 ||x||∞

= max (1-i-m) SUM^n_j=1: |aij|
=max row sum

Question 4

||A||2=

Answer 4

max ||Ax||2
x=!0 ||x||2

= ^2root [λmax(A∗A)]
= spectral norm
λmax denotes the largest eigenvalue

Question 5

Let A ∈ C^n×n
Then
lim A^k = 0 ⇔
k→∞

Answer 5

ρ(A) < 1

Question 6

Gershgorin’s theorem

Answer 6

The eigenvalues of A ∈ C^n×n
Lie in the union of the n discs in the complex plane

Di = { z ∈ C : |z − aii| ≤ SUM^n_j=1,j=!i: |aij| }
i=1,2,..n