Talk:Analytic function of a matrix

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

Pleeeeeeeeease

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)

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

Proof

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

General Taylor series

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 $A(\eta )=A+\eta B$ and I want to expand out $A(\eta )^{3}$ in terms of $\eta$. 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.

$f(a+\epsilon b)=f(a)+f'(a){\frac {\eta b}{1!}}+f''(a){\frac {\eta ^{2}b^{2}}{2!}}+f'''(a){\frac {\eta ^{3}b^{3}}{3!}}=a^{3}+3a^{2}(\eta b)+3a(\eta b)^{2}+(\eta b)^{3}$

Obviously this is just the expansion of $(a+\eta b)^{3}$ for scalars. We sub in the matrices to get

$f(A+\epsilon B)=A^{3}+3A^{2}(\eta B)+3A(\eta B)^{2}+(\eta B)^{3}$

Now expand $A(\eta )^{3}$ directly,

$(A+\eta B)^{3}=(A+\eta B)(A^{2}+\eta AB+\eta BA+\eta ^{2}B^{2})=A^{3}+\eta (A^{2}B+ABA+BA^{2})+\eta ^{2}(AB^{2}+BAB+B^{2}A)+\eta ^{3}B^{3}$

The expressions don't match unless $[A,B]=0$ (assuming A,B non-zero etc). A similar thing happens with $x^{-1}$ for $x=A+\eta B$ about $\eta =0$. Converting from the scalar expression gives terms of the form $(A^{-1})^{n}B^{n}$ while the expansion is actually in terms of $(A^{-1}B)^{n}$. — 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)

Functional matrix?

       (       )
       ( 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)

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)

Matrix-vector product

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