norms

Question 1

vector norm:

Answer 1

a function ||.||:C^(n)->R satisfying:
||x||>=0 with equality iff x=0 (positive definiteness)
||αx||=|α|||x|| for all α in C and x in C^n (absolute homogeneity)
||x+y||<=||x||+||y|| for all x,y in C^n (triangle inequality)

Question 2

manhattan norm:

Answer 2

aka taxi cab norm, ||x||1=|x1|+…+|xn|=(n)Σ(i=1)|xi|

Question 3

euclidean length:

Answer 3

||x||2=(|x1|^(2)+…+|xn|^(2))^(1/2)=((n)Σ(i=1)|xi|^2)^(1/2)=(x*x)^(1/2)

Question 4

maximum norm:

Answer 4

||x||∞=max|xi|

Question 5

p-norm:

Answer 5

||x||p=(|x1|^(p)+…+|xn|^(p))^(1/p)=((n)Σ(i=1)|xi|^p)^(1/p), p>=1, the manhattan norm euclidean length and maximum norm are all special cases of this

Question 6

hölder inequality:

Answer 6

|x*y|<=||x||p ||y||q, (1/p)+(1/q)=1

Question 7

cauchy-schwarz inequality:

Answer 7

|x*y|<=||x||2 ||y||2, special case of the hölder inequality where p=q=2

Question 8

equivalent:

Answer 8

all norms on C^n are equivalent, meaning if ||.||α and ||.||β are norms on C^n then there exist positive constants δ1 and δ2 such that δ1||x||α<=||x||β<=δ2||x||α for all x in C^n

Question 9

vector norms and convergence:

Answer 9

a sequence {v^k} of vectors in C^n converges to a vector v with respect to the vector norm ||.|| iff lim(k->∞)(||v^(k)-v||=0), choice of norm doesn’t matter cause they’re equivalent

Question 10

matrix norm:

Answer 10

a function ||.||:C^(mxn)->R that satisfies the matrix versions of the vector norm properties (just swap x with A and y with B or smth)

Question 11

frobenius norm:

Answer 11

||A||F=((m)Σ(i=1)(m)Σ(j=1)|aij|^2)^(1/2)=(trace(A*A))^(1/2)

Question 12

subordinate:

Answer 12

👀
some matrix norms are subordinate to (aka induced by) vector norms. given a vector norm, the corresponding subordinate vector norm is defined by ||A||=max(||Ax||/||x||), x!=0

Question 13

matrix p-norm:

Answer 13

||A||p=max(||Ax||p/||x||p), x!=0

Question 14

max column sum:

Answer 14

||A||1=max(||Ax||1/||x||1)=max((m)Σ(i=1)|aij|)

Question 15

max row sum:

Answer 15

||A||∞=max(||Ax||∞/||x||∞)=max((n)Σ(j=1)|aij|)=||A^T||1

Question 16

spectral norm:

Answer 16

||A||2=max(||Ax||2/||x||2)=root(λ(max)(A*A)) where λ(max) denotes the largest eigenvalue

Question 17

consistent:

Answer 17

a norm is consistent if it satisfies ||AB||<=||A||||B|| when AB is defined, the frobenius and all subordinate norms are consistent

Question 18

mixed norm:

Answer 18

||A||∞,1:=max(||Ax||∞/||x||1)=max(aij), not consistent

Question 19

spectral radius:

Answer 19

ρ(A)=max{|λ|: λ is an eigenvalue of A} (biggest eigenvalue. basically)

Question 20

consistent matrix norms and the spectral radius:

Answer 20

for any consistent matrix norm, ρ(A)<=||A||

Question 21

limits and the spectral radius:

Answer 21

let A be a square matrix, lim(k->∞)A^k=0 <=> ρ(A)<1

Question 22

gershgorin’s theorem:

Answer 22

the eigenvalues of A lie in the union of the n discs in the complex plane Di={z in C:|z-aii|<=(n)Σ(j=1, j!=i)|aij|}