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functional calculus for Hermitian matrices (Definition)

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)$.



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See Also: functional calculus
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Cross-references: permutation, well-defined, diagonal, diagonal matrix, orthogonal matrix, diagonalize, spectral theorem, contained, eigenvalues, matrix, Hermitian, symmetric, function, interval, real
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This is version 1 of functional calculus for Hermitian matrices, born on 2004-10-02.
Object id is 6271, canonical name is FunctionalCalculusForHermitianMatrices.
Accessed 1408 times total.

Classification:
AMS MSC: 47C05 (Operator theory :: Individual linear operators as elements of algebraic systems :: Operators in algebras)
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