Analytic function of a matrix: Difference between revisions

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===Diagonalizable matrices===

IfA thesquare matrix {{mvar|A}} is [[diagonalizable matrix|diagonalizable]], theif problemthere mayis bean reduced[[invertible tomatrix]] an{{mvar|P}} arraysuch ofthat the<math>D function= onP^{-1}\,A\,P</math> eachis a [[diagonal matrix]], that is, {{math|D}} has the eigenvalue.shape

:<math>f(A) D= P \begin{bmatrix}

This is to say we can find a matrix {{mvar|P}} and a [[diagonal matrix]] {{mvar|D}}

d_1 & \cdots & 0 \\

such that <math>A = P~ D~ P^{-1}</math>.

\vdots & \ddots & \vdots \\

Applying the power series definition to this decomposition, we find that {{math|''f''&hairsp;(''A'')}} is defined by

0 & \cdots & d_n

:<math>f(A) = P \begin{bmatrix}

\end{bmatrix}.</math>

such thatAs <math>A = P~ \,D~ \,P^{-1},</math>. it is natural to set

:<math>f(A)=P\, \begin{bmatrix}

f(d_1) & \cdots & 0 \\

\vdots & \ddots & \vdots \\

0 & \cdots & f(d_n)

\end{bmatrix} \,P^{-1} ~, .</math>

where <math>d_1, \dots, d_n</math> denote the diagonal entries of ''D''.

It can be verified that the matrix {{math|''f''(''A'')}} does not depend on a particular choice of {{mvar|P}}.

For example, suppose one is seeking <math>\Gamma(A) = (A-1)!</math> for

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1&3\\

2&1

\end{bmatrix} ~. </math>

One has