# norm

## Purpose

Vector and matrix norms.

## Synopsis

````n = norm(X)`
`n = norm(X,p)`
n = norm(X,'fro')
```

## Description

The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. Several different types of norms can be calculated:

`n = norm(X)`, where `X` is a matrix, is the largest singular value of `X`.

`n = norm(X,p)`, lets you specify a value to indicate largest singular value, largest column sum, or largest row sum of matrix `X`:

• `norm(X,1)` is the 1-norm, or largest column sum of `X`, `max(sum(abs((X)))`.
• `norm(X,2)` is the same as `norm(X)`.
• `norm(X,inf)` is the infinity norm, or largest row sum of `X`, `max(sum(abs(X')))`.
`n = norm(X,'fro')` is the F-norm of matrix `X`, `sqrt(sum(diag(X'`*`X)))`.

When the `X` is a vector, slightly different rules apply:

• `norm(x,p) = sum(abs(x).^p)^(1/p)`.
• `norm(x) = norm(x,2)`.
• `norm(x)/sqrt(n)` is the root-mean-square (RMS) value.
• `norm(x,inf) = max(abs(x))`.
• `norm(x, -inf) = min(abs(x))`.

## Examples

For a matrix

````A =`
`    1    2    3`
`    4    5    6`
`    7    8    9`
`           `
`norm(A)       = 16.8481`
`norm(A,1)     = 18`
`norm(A,2)     = 16.8481`
`norm(A,inf)   = 24`
`norm(A,'fro') = 16.8819`
```
For a vector

````v =`
`    1    2    3`
`          `
`norm(v)       = 3.7417`
`norm(v,1)     = 6`
`norm(v,2)     = 3.7417`
`norm(v,Inf)   = 3`
`norm(v,pi)    = 3.2704`
```

````cond`, `max`, `min`, `rcond`, `svd`