functional calculus for Hermitian matrices
Let be a real interval, a real-valued function on , and let be an real symmetric (and thus Hermitian) matrix whose eigenvalues are contained in .
By the spectral theorem, we can diagonalize by an orthogonal matrix , so we can write where is the diagonal matrix consisting of the eigenvalues . We then define
where denotes the diagonal matrix whose diagonal entries are given by .
It is easy to verify that is well-defined, i.e. a permutation of the eigenvalues corresponds to the same definition of .
Title | functional calculus for Hermitian matrices |
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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 |