norm (MATLAB Function Reference)

Vector and matrix norms

Syntax

n = norm(A)
n = norm(A,p)

Description

The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. The norm function calculates several different types of matrix norms:

n = norm(A) returns the largest singular value of A, max(svd(A)).

n = norm(A,p) returns a different kind of norm, depending on the value of p:

If p is...
Then norm returns...

1
The 1-norm, or largest column sum of A, max(sum(abs((A))).

2
The largest singular value (same as norm(A)).

inf
The infinity norm, or largest row sum of A, max(sum(abs(A'))).

'fro'
The Frobenius-norm of matrix A, sqrt(sum(diag(A'*A))).

If `p` is...	Then `norm` returns...
`1`	The 1-norm, or largest column sum of `A`, `max(sum(abs((A)))`.
`2`	The largest singular value (same as `norm(A)`).
`inf`	The infinity norm, or largest row sum of `A`, `max(sum(abs(A')))`.
`'fro'`	The Frobenius-norm of matrix `A`, `sqrt(sum(diag(A'`*`A)))`.

When A is a vector, slightly different rules apply:

norm(A,p)
Returns sum(abs(A).^p)^(1/p), for any 1 p .

norm(A)
Returns norm(A,2).

norm(A,inf)
Returns max(abs(A)).

norm(A,-inf)
Returns min(abs(A)).

`norm(A,p)`	Returns `sum(abs(A).^p)^(1/p)`, for any 1 p .
`norm(A)`	Returns `norm(A,2)`.
`norm(A,inf)`	Returns `max(abs(A))`.
`norm(A,-inf)`	Returns `min(abs(A))`.

Remarks

To obtain the root-mean-square (RMS) value, use norm(A)/sqrt(n). Note that norm(A), where A is an n-element vector, is the length of A.

See Also

cond, condest, normest, rcond, svd

nonzeros normest