# functional calculus for Hermitian matrices

Let $I\subset\mathbb{R}$ 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 $M=ODO^{-1}$ where $D$ is the diagonal matrix consisting of the eigenvalues $\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}$. We then define

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

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 $f(A)$.

Title functional calculus for Hermitian matrices FunctionalCalculusForHermitianMatrices 2013-03-22 14:40:12 2013-03-22 14:40:12 mathcam (2727) mathcam (2727) 4 mathcam (2727) Definition msc 47C05 FunctionalCalculus