functional calculus for Hermitian matrices

Let I be a real interval, f a real-valued function on I, and let M be an n×n real symmetricPlanetmathPlanetmath (and thus Hermitian) matrix whose eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath are contained in I.

By the spectral theoremMathworldPlanetmathPlanetmath, we can diagonalize M by an orthogonal matrixMathworldPlanetmath O, so we can write M=ODO-1 where D is the diagonal matrixMathworldPlanetmath consisting of the eigenvalues {λ1,λ2,,λn}. We then define


where f(D) denotes the diagonal matrix whose diagonal entries are given by f(λi).

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

Title functional calculus for Hermitian matrices
Canonical name FunctionalCalculusForHermitianMatrices
Date of creation 2013-03-22 14:40:12
Last modified on 2013-03-22 14:40:12
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 4
Author mathcam (2727)
Entry type Definition
Classification msc 47C05
Related topic FunctionalCalculus