Lecture 10

Question 1

Norm properties

Answer 1

||x||: R^n > R0+ norm if

1. ||x|| > 0 iff x≠0
2. ||γx|| = |γ| ||x||
3. ||x+y|| ≤ ||x|| + ||y||

Question 2

p-norms (1, 2, ∞)

Answer 2

||(x1 xn)||_p = sqrt^p(|x1|^p + … + |xn|^p)

||x||_1 = |x1| + ... + |xn|
||x||_2 = sqrt(x1^2 + ... + xn^2)
||x||_∞ = max_i |xi|

Question 3

Unit balls (1, 2, ∞)

Answer 3

circle, diamond, square

Question 4

If p<1?

Answer 4

unit ball not convex, triangle inequality contradicted

Question 5

Distance between x and y

Answer 5

||x-y||

Question 6

Magnitude of error of a vector

Answer 6

||trueval - appval||

Question 7

abs/rel errors of approximating (40.114, -88.224) as (40,-88)?

Answer 7

abs = ||x-^x||_2 = 0.2513
rel = ||x-^x||_2 / ||x||_2 = 2.59e-3

Question 8

Frobenius norm

Answer 8

Matrix norm, flatten and use vector norm:

||A||_F = sqrt(sum_i,j aij^2)

Question 9

Induced matrix norms (general)

Answer 9

maximum amplification of the norm of any vector multiplied by the matrix
||A|| = max_{||x||≠0} ||Ax||/||x||
= max_{||y||=1} ||Ay||

Question 10

Induced matrix norms (1, 2, ∞)

Answer 10

||A||_1 = maxj sum(i:1>n) |Aij| (max of abs column sum)
||A||_∞ = maxi sum(j:1>n) |Aij| (max abs row sum)
||A||_2 = maxk σ_k (singular value)

Question 11

Matrix norms properties

Answer 11

vector norm properties + submultiplicativity:

1. ||Ax|| ≤ ||A|| ||x||
2. ||AB|| ≤ ||A|| ||B||

Question 12

∞ norm:
1 2
3 4

Answer 12

7

Question 13

1 norm:
1 2
3 4

Answer 13

6

Question 14

You are given for A, 3 vectors x, y, z. ||x||=2, ||y||=1, ||z||=3 and ||Ax||=20, ||Ay||=5, ||Az||=90, what’s the largest lower bound for ||A|| you can derive?

Answer 14

max(||Ax||/||x||, …) = 30

Question 15

2 norm:
100 0 0
0 13 0
0 0 0.5

Answer 15

100

Question 16

Inverse of a diagonal matrix
d1 0 0
0 d2 0
0 0 d3

Answer 16

1/d1 0 0
0 1/d2 0
0 0 1/d3

Question 17

inner product

Answer 17

if u,v,w in V:
1. positivity <u> ≥ 0
2. definiteness <u>=0 iff u=0
3. symmetric <u> = 
3. linearity  = ɑ<u> + β
Inner product represents similarity between two vectors, <u> = 0 if orthogonal
algebraic: a.b = ab.T
geometric: a.b = ||a|| ||b|| cosθ</u></u></u></u></u>

Question 18

Relative error wrt 1-norm of with fps representation 1.s1s2s3 x 2^m:

1. 71875
2. 7500
3. 0000

Answer 18

0.014492

Question 19

python norm

Answer 19

np. linalg.norm(x, np.inf)

np. linalg.norm(x, 2)…