Algebraic element

Spectrum of an algebraic element

Let 𝐴 be a K-monoid over 𝕂 and 𝑎 ∈𝐴 be an algebraic element with minimal polynomial 𝑚𝑎(𝑥) ∈𝕂[𝑥]. The roots of 𝑚𝑎(𝑥) are called the eigenvalues of 𝑎, and the set of all eigenvalues falg

Spec⁡(𝑎)={𝜆∈𝕂:𝑚𝑎(𝜆)=0}

is called the spectrum of 𝑎.¹ Clearly 𝑎 invertible iff 0 ∉Spec⁡(𝑎).

Properties

• Spectral mapping theorem

In addition

1. Spec⁡(𝑎𝑏) =Spec⁡(𝑏𝑎)²

Proof

proof

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Footnotes

1. 2008. Advanced Linear Algebra, §18, p. 461 ↩

2. Stated with partial proof in 2008. Advanced Linear Algebra, §18, p. 462 ↩