norm

Arguments

x

vector or matrix of real or complex numbers (full or sparse storage)

normType

• For a matrix x: a number among 1, 2, %inf, -%inf, or a word among "inf" (or "i") or "fro" (or "f").
• For a vector x: any number or %inf, -%inf; or a word "inf" ("i"), "fro" ("f").

Default value = 2.

y

norm: single positive real number.

Description

For matrices

norm(x)

or norm(x,2) is the largest singular value of x (max(svd(x))).

norm(x,1)

The l_1 norm x (the largest column sum : max(sum(abs(x),'r')) ).

norm(x,'inf'),norm(x,%inf)

The infinity norm of x (the largest row sum : max(sum(abs(x),'c')) ).

norm(x,'fro')

Frobenius norm i.e. sqrt(sum(diag(x'*x))).

For vectors

norm(v,p)

The l_p norm sum(abs(v(i))^p)^(1/p) .

norm(v), norm(v,2)

The l_2 norm

norm(v,'inf')

max(abs(v(i))).

Remark

norm([]) and norm([],p) return 0.

norm(x) and norm(x,p) return NaN if x contains NaNs.

Examples

A = [1,2,3];
norm(A,1)
norm(A,'inf')
A = [1,2;3,4]
max(svd(A)) - norm(A)

A = sparse([1 0 0 33 -1])
norm(A)

See also

• h_norm — H-infinity norm
• dhnorm — discrete H-infinity norm
• h2norm — H2 norm of a continuous time proper dynamical system
• abs — absolute value, magnitude
• svd — singular value decomposition