Spectrum of an algebraic element

Let $𝐴$ be a $𝕂$ -monoid over $𝕂$ and $𝑎 \in 𝐴$ be an algebraic element with minimal polynomial $𝑚 𝑎 (𝑥) \in 𝕂 [𝑥]$ . The roots of $𝑚 𝑎 (𝑥)$ are called the eigenvalues of $𝑎$ , and the set of all eigenvalues #m/def/ralg

S p e c (𝑎) = {𝜆 \in 𝕂 : 𝑚 𝑎 (𝜆) = 0}

is called the spectrum of $𝑎$ .¹ Clearly $𝑎$ invertible iff $0 \notin S p e c (𝑎)$ .

Properties

Spectral mapping theorem

In addition

$S p e c (𝑎 𝑏) = S p e c (𝑏 𝑎)$ ²

Proof

#missing/proof

#state/tidy | #lang/en | #SemBr

2008. Advanced Linear Algebra, §18, p. 461 ↩
Stated with partial proof in 2008. Advanced Linear Algebra, §18, p. 462 ↩

Spectrum of an algebraic element

Properties

Footnotes