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)