Application of matrix fuctions Given some $f:\mathbb{R} \rightarrow \mathbb{R}$ with $f(x) = \sum_{i=1}^{n}a_ix^i$ it is possible to define $f(A) = \sum_{i=0}^{\infty}a_iA^i$ that converges under some conditions.
A classical application for this is for linear systems of ODE's. If $x'(t) = Ax(t)$ then $x(t) = e^{At}x_o$.
My question is: are there other applications that make use of some matrix function? (not necessarily $e^A$, but also $\sqrt{A}$, $\ln(A)$ etc.
Thanks! 
 A: For the record, the idea of applying functions to matrices/operators in this way is called functional calculus.
Some important examples: 


*

*$f(x) = \sqrt{x}$ is very important in the study of positive semi-definite matrices/operators.  An absolutely essential fact to the study of positive semi-definite matrices is that if $A$ is a bounded positive operator, then there is a unique positive $\sqrt{A}$ satisfying $(\sqrt{A})^2 = A$, which necessarily commutes with $A$.  More generally, one needs the square root function in order to express the polar decomposition of a matrix, wherein any matrix $A$ can be written in the form
$$
A = U\sqrt{A^*A}
$$
for some unitary (orthogonal) $U$.  (Note: $A^*$ is the "conjugate transpose" of $A$, which is just the transpose when $A$ is real).

*$f(x) = \ln(x)$ is very important to the study of Lie groups and Lie algebras.  Both of these may be thought of in terms of matrices... most of the time.  Importantly, the function $e^x$ takes us from the Lie Algebra to the corresponding Lie Group, while $\ln(x)$ takes us from the Lie Group to the corresponding Lie Algebra.  In a sense, we can think of this as an extension of the usual ODE application of $e^x$.

*$f(x) = 1/x$ (or more typically, $f(x) = 1/(\lambda - x)$) comes up a lot.  A very common (and useful!) trick is to note that whenever the series for $f(x) = 1/x$ converges, the matrix $f(A)$ must be the inverse of $A$, which means that $A$ has to be invertible.  In matrix/operator perturbation theory, it is often helpful to characterize the eigenvalues of an operator $T$ using contour integrals in the complex plane of the form
$\int_\Gamma \frac{1}{\xi - T} d\xi = \int_{\Gamma} (\xi I - T)^{-1} d\xi$.
