Editing Jordan normal form (section)

== Matrix functions ==
{{Main|Matrix function}}
Iteration of the Jordan chain motivates various extensions to more abstract settings. For finite matrices, one gets matrix functions; this can be extended to compact operators and the holomorphic functional calculus, as described further below.

The Jordan normal form is the most convenient for computation of the matrix functions (though it may be not the best choice for computer computations). Let ''f''(''z'') be an analytical function of a complex argument. Applying the function on a ''n''×''n'' Jordan block ''J'' with eigenvalue ''λ'' results in an upper triangular matrix:

:<math>
f(J)
=\begin{bmatrix}
 f(\lambda) & f'(\lambda) & \tfrac{f''(\lambda)}{2} & \cdots   &  \tfrac{f^{(n-1)}(\lambda)}{(n-1)!}\\
 0  & f(\lambda) & f'(\lambda) & \cdots   & \tfrac{f^{(n-2)}(\lambda)}{(n-2)!} \\
 \vdots  & \vdots  & \ddots & \ddots   & \vdots \\ 
 0  & 0  & 0  & f(\lambda) & f'(\lambda) \\
 0  & 0  & 0  & 0   & f(\lambda)
\end{bmatrix},</math>

so that the elements of the ''k''-th superdiagonal of the resulting matrix are <math>\tfrac{f^{(k)}(\lambda)}{k!}</math>. For a matrix of general Jordan normal form the above expression shall be applied to each Jordan block.

The following example shows the application to the power function ''f''(''z'')&nbsp;=&nbsp;''z<sup>n</sup>'':
:<math>
\begin{bmatrix}
 \lambda_1 & 1 & 0 & 0 & 0 \\
 0 & \lambda_1 & 1 & 0 & 0 \\
 0 & 0 & \lambda_1 & 0 & 0 \\ 
 0 & 0 & 0 & \lambda_2 & 1 \\
 0 & 0 & 0 & 0 & \lambda_2
\end{bmatrix}^n
=\begin{bmatrix}
 \lambda_1^n & \tbinom{n}{1}\lambda_1^{n-1} & \tbinom{n}{2}\lambda_1^{n-2} & 0   & 0 \\
 0  & \lambda_1^n & \tbinom{n}{1}\lambda_1^{n-1} & 0   & 0 \\
 0  & 0  & \lambda_1^n & 0   & 0 \\ 
 0  & 0  & 0  & \lambda_2^n & \tbinom{n}{1}\lambda_2^{n-1} \\
 0  & 0  & 0  & 0   & \lambda_2^n
\end{bmatrix},</math>
where the binomial coefficients are defined as <math display="inline">\binom{n}{k}=\prod_{i=1}^k \frac{n+1-i}{i}</math>. For integer positive ''n'' it reduces to standard definition
of the coefficients. For negative ''n'' the identity <math display="inline">\binom{-n} k = (-1)^k\binom{n+k-1}{k}</math> may be of use.