Talk:Analytic function of a matrix

Latest comment: 4 years ago by HerrHartmuth in topic Matrix-vector product

What is the source? Seems to be [Higham "Functions of matrices: Theory and computation"].

Pleeeeeeeeease

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Please some one put a few examples of matrix functions here that are solved and computed. —Preceding unsigned comment added by 128.100.86.53 (talk) 21:50, 3 February 2010 (UTC)Reply

OK, did. Cuzkatzimhut (talk) 15:40, 3 February 2016 (UTC)Reply

Proof

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There should be a derivation and proof of the Matrix function equation f(A) = ...

General Taylor series

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The example given is a Maclaurin series, not the general Taylor series. There's an important difference because you can't just "Put in the matrix for x" for the general expansion. For example, suppose I have   and I want to expand out   in terms of  . We can compute this in 2 ways, brute force expansion or taking the scalar Taylor series and putting in the relevant matrices, in the same manner as the example on the page.

 

Obviously this is just the expansion of   for scalars. We sub in the matrices to get

 

Now expand   directly,

 

The expressions don't match unless   (assuming A,B non-zero etc). A similar thing happens with   for   about  . Converting from the scalar expression gives terms of the form   while the expansion is actually in terms of  . — Preceding unsigned comment added by 94.174.139.32 (talk) 08:32, 8 July 2012 (UTC) <-- Sorry that was me AlphaNumeric (talk) 19:04, 8 July 2012 (UTC)Reply

Functional matrix?

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       (       )
       ( a * x )
f(x) = (       )
       ( b * x )
       (       )

How is this matrix called? (I would call it functional matrix but I didn't find any info about it)

It could also be extended:

         (       )
         ( a * x )
f(x,y) = (       )
         ( b * y )
         (       )

....or with any other matrix features like scalars, aso. Any ideas? --178.197.224.240 (talk) 18:12, 28 December 2013 (UTC)Reply

Got it, above matrizes could also be written like this (never thought about scalars that way):

       (   )
       ( a )
f(x) = (   ) * x
       ( b )
       (   )

And:

          (   )   (   )
          ( a )   ( x )
f(x, y) = (   ) * (   )
          ( b )   ( y )
          (   )   (   )

But do they have a distinct name? --178.197.224.240 (talk) 18:38, 28 December 2013 (UTC)Reply

Matrix-vector product

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Often one does not need the explicit computation of the matrix function   but rather its action on a vector  . For example for an iterative solver such as GMRES. HerrHartmuth (talk) 09:03, 6 December 2019 (UTC)Reply