Chapter 1

Question 1

Norm

Answer 1

A real number ||x|| such that:

i) ||x||>0 , for all x=/=0
ii) ||αx||=|α| ||x||
iii) ||x+y||<=||x|| + ||y||

Question 2

Subordinate Matrix Norm

Answer 2

||A|| = max (||Ax||/||x||)

i) ||A|| > 0, for all A=/=0
ii) ||αA|| = |α| ||A||
iii) ||Ax|| <= ||A|| ||x||
iv) ||AB|| <= ||A|| ||B||
v) ||A + B|| <= ||A|| + ||B||

Question 3

Cauchy-Schwartz Inequality

Answer 3

² <=

Question 4

P-norm

Answer 4

||x|| = (∑|xi|^p)^(1/p)

Question 5

Infinity - norm

Answer 5

||x|| = max (|xi|)

Question 6

Spectral radius

Answer 6

p(A) = max (λ)

p(A) <= ||A||

Question 7

Estimating B inverse

Answer 7

<= 1/(1-||A||)

where B = I + A and ||A||<1

Question 8

Error in x when the system is changed by δb

Answer 8

<= K(A)||δb||/||b||

Question 9

Condition Number

Answer 9

K(A)=||A|| ||A-1||
K(A) >= 1
K(A) >= |λ1|/|λn|
K(αA) = K(A)

Question 10

Matrix norm subordinate to the infinity norm

Answer 10

Maximum row sum

Question 11

Matrix norm subordinate to the L1 normed vector space

Answer 11

Maximum column sum

Question 12

||x+y||² <= ||x||² + ||y||²

Answer 12

||x+y||² 
= 
=  + 2 + 
<=  +  + 2sqrt() by C-S
<=  +
= ||x||² + ||y||²