Re: Euclidean Matrix Norm
- To: mathgroup at smc.vnet.net
- Subject: [mg5351] Re: Euclidean Matrix Norm
- From: Daniel Lichtblau <danl>
- Date: Wed, 27 Nov 1996 01:48:04 -0500
- Organization: wolfram.com
- Sender: owner-wri-mathgroup at wolfram.com
Armin Gerritsen wrote:
>
> Does anyone know if Mathematica (2.2.2) can calculate the Euclidean
> matrix-norm of a matrix. (== max |Ax|/|A| , where x a non-zero vector and A
> a square matrix).
>
> Tanks,
>
> A.A.Gerritsen
The Euclidean norm is given by the largest singular value. You can get
this as
First[SingularValues[matrix][[2]]]
because (i) the list of singular values is the second component in the
output of SingularValues (the first and third being the pre- and
post-multiplier matrices; see usage message below), and (ii) the
singular values are ordered by decreasing magnitude.
In[12]:= ??SingularValues
SingularValues[m] gives the singular value decomposition for a numerical
matrix m. The result is a list {u, w, v}, where w is the list of
singular
values, and m can be written as
Conjugate[Transpose[u]].DiagonalMatrix[w].v.
Attributes[SingularValues] = {Protected}
Options[SingularValues] = {Tolerance -> Automatic}
Daniel Lichtblau
Wolfram Research
danl at wolfram.com