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The spectral functional calculus f(T) for a normal endomorphism

Definition

Let $V$ be a finite-dimensional complex inner product space, let $T : V \to V$ be normal, and write the spectral resolution from A normal endomorphism is a sum of its eigenvalues times pairwise orthogonal projections, and each spectral projection is a polynomial in the endomorphism as

$T = \sum_{j = 1}^{r} λ_{j} P_{j},$

where the $λ_{j}$ are the distinct eigenvalues of $T$ and $P_{j}$ is the orthogonal projection onto $E_{λ_{j}} (T)$ . For a function $f : σ (T) \to C$ on the spectrum of $T$ , define

$f (T) : = \sum_{j = 1}^{r} f (λ_{j}) P_{j} .$

This is the spectral functional calculus of $T$ . For the identity function ${id}_{σ (T)}$ , one has ${id}_{σ (T)} (T) = T$ .

Depends on

A normal endomorphism is a sum of its eigenvalues times pairwise orthogonal projections, and each spectral projection is a polynomial in the endomorphism

Used by

For normal endomorphisms, the spectral functional calculus respects sums, products, adjoints, and composition of scalar functions Proposition

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)