functional calculus for Hermitian matrices

Let $I\subset ℝ$ be a real interval, $f$ a real-valued function on $I$, and let $M$ be an $n\times n$ real symmetric (and thus Hermitian) matrix whose eigenvalues are contained in $I$.

By the spectral theorem, we can diagonalize $M$ by an orthogonal matrix $O$, so we can write $M=OD{O}^{-1}$ where $D$ is the diagonal matrix consisting of the eigenvalues $\{{\lambda }_1,{\lambda }_2,\mathrm{\dots },{\lambda }_n\}$. We then define

$f(A)=Of(D)O^{-1},$

where $f(D)$ denotes the diagonal matrix whose diagonal entries are given by $f({\lambda }_i)$.

It is easy to verify that $f(A)$ is well-defined, i.e. a permutation of the eigenvalues corresponds to the same definition of $f(A)$.