Talk:Analytic function of a matrix

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

Pleeeeeeeeease

Latest comment: 10 years ago2 comments2 people in discussion

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

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

General Taylor series

Latest comment: 14 years ago1 comment1 person in discussion

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 $A (η) = A + η B$ and I want to expand out $A (η)^{3}$ in terms of $η$ . 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 + ϵ b) = f (a) + f^{'} (a) \frac{η b}{1!} + f^{″} (a) \frac{η^{2} b^{2}}{2!} + f^{‴} (a) \frac{η^{3} b^{3}}{3!} = a^{3} + 3 a^{2} (η b) + 3 a (η b)^{2} + (η b)^{3}

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

f (A + ϵ B) = A^{3} + 3 A^{2} (η B) + 3 A (η B)^{2} + (η B)^{3}

Now expand $A (η)^{3}$ directly,

(A + η B)^{3} = (A + η B) (A^{2} + η A B + η B A + η^{2} B^{2}) = A^{3} + η (A^{2} B + A B A + B A^{2}) + η^{2} (A B^{2} + B A B + B^{2} A) + η^{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 $x = A + η B$ about $η = 0$ . 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)Reply

Functional matrix?

Latest comment: 12 years ago2 comments1 person in discussion

       (       )
       ( 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

Latest comment: 6 years ago1 comment1 person in discussion

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)Reply