spectrum of A-μI

Let A be an endomorphismPlanetmathPlanetmath of the vector spaceMathworldPlanetmath V over a field k. Denote by σ(A) the spectrum of A. Then we have:

Theorem 1.

Theorem 1 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to:

Theorem 2.

λ is a spectral value of A if and only if λ-μ is a spectral value of A-μI.

Proof of Theorem 2.

Note that


and thus A-λI is invertiblePlanetmathPlanetmathPlanetmathPlanetmath if and only if (A-μI)-(λ-μ)I is invertible. Equivalently, λ is a spectral value of A iff λ-μ is a spectral value of (A-μI), as desired. ∎

Title spectrum of A-μI
Canonical name SpectrumOfAmuI
Date of creation 2013-03-22 15:32:49
Last modified on 2013-03-22 15:32:49
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 9
Author PrimeFan (13766)
Entry type Theorem
Classification msc 15A18
Related topic SpectralValuesClassification