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Vector And Matrix Norms
Syntax norm( x )
norm( 
xp )
See Also norm operator abs , max

Description
Returns the norm of x where x is an integer, real, double-precision, or complex matrix. If x is integer or real, the return value is a real scalar, otherwise the return value is a double-precision scalar. If the argument p is present, it can be a positive integer, real, or double-precision scalar. It can also be the character vectors "fro", "inf", or "-inf".

If x is an empty matrix , the return value is equal to zero.

Scalars
In all cases, the norm of a scalar is equal to its absolute value. If you enter
     norm(-5)
O-Matrix will respond
     5

Vectors

Frobenius Norm
If x is a vector and p is equal to "fro", the Frobenius norm of x is computed. The Frobenius norm of x is the square root of the sum of the squares of the absolute value of the elements of x. For example
     norm([3, 4], "fro")
will result in
     5

Maximum Norm
If x is a vector and p is equal to "inf" the return value is the maximum of the absolute value of the elements of x. For example
     norm([3, 4], "inf")
will result in
     4

Minimum Norm
If x is a vector and p is equal to "-inf" the return value is the minimum of the absolute value of the elements of x. For example
     norm([3, 4], "-inf")
will result in
     3

P-Norm
If x is a vector and p is a number other than plus or minus infinity, the return value is equal to
     /  p    p          p \ (1/p)
     | x  + x  + ... + x  |
     \  1    2          n /

Matrices

Euclidian Norm
If x is not a vector or an empty matrix and p is not present or it is equal to 2, the return value is the maximum absolute singular value corresponding to the matrix x. For example,
     x = { ...
          [3, 0], ...
          [0, 4] ...
     } 
     norm(x)
will result in
     4

Frobenius Norm
If x is not a vector or an empty matrix and p is equal to "fro", the Frobenius norm of x is computed. The Frobenius norm of x is the square root of the sum of the squares of the absolute value of the elements of x. For example
     x = { ...
          [3, 0], ...
          [0, 4] ...
     } 
     norm(x, "fro")
will result in
     5

One Norm
If x is not a vector or an empty matrix and p is equal to 1, the return value is the maximum with respect to the columns of x of the sum of the absolute value of the elements in each column. For example,
     x = { ...
          [1, 2], ...
          [3, 4] ...
     } 
     norm(x, 1)
will result in
     6

Infinity Norm
If x is not a vector or an empty matrix and p is equal to "inf", the return value is the maximum with respect to the rows of x of the sum of the absolute value of the elements in each row. For example,
     x = { ...
          [1, 2], ...
          [3, 4] ...
     } 
     norm(x, "inf")
will result in
     7

Reference
The values plus and minus infinity can be used in place of "inf" and "-inf" respectively.