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Spectrum in a finite-dimensional matrix algebra

Example

Let $n \geq 1$ and let $M_{n} (C)$ carry the Euclidean operator norm induced by identifying $M_{n} (C)$ with $B (C^{n})$ , $C^{n}$ having the Euclidean norm, and by the operator norm on that space (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Then $M_{n} (C)$ is a unital complex Banach algebra (Unital Banach algebra), and for every $A \in M_{n} (C)$

$σ (A) = {λ \in C : \det (λ I - A) = 0}$

with the spectrum taken in that algebra (Spectrum and resolvent set in a Banach algebra) and the determinant of For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix.

Facts & Assumptions

Given: An integer $n \geq 1$ , the algebra $M_{n} (C)$ of $n \times n$ complex matrices with the operator norm, and a matrix $A \in M_{n} (C)$ .

[L1]

The operator norm is submultiplicative and $∥ I ∥ = 1$ ; an element is invertible in $M_{n} (C)$ exactly when it has a two-sided inverse matrix, and $λ \in σ (A)$ exactly when $λ I - A$ is not invertible (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Spectrum and resolvent set in a Banach algebra, Unital Banach algebra).

[L2]

The adjugate identity: $B a d j (B) = a d j (B) B = \det (B) I$ for every $B \in M_{n} (C)$ (For every positive-sized square matrix over a commutative ring, $A adj (A) = adj (A) A = \det (A) I$ , For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L3]

Determinants are multiplicative: $\det (B C) = \det (B) \det (C)$ (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ).

[L4]

Finite-dimensional normed spaces are complete, so $M_{n} (C) = B (C^{n})$ is complete for the operator norm (Every finite-dimensional normed space is Banach).

Verification

technique · direct

1.1

$M_{n} (C)$ is an associative complex algebra under matrix multiplication with unit $I$ ; by [L1] the operator norm is submultiplicative with $∥ I ∥ = 1$ , and by [L4] the space is complete; hence it is a unital complex Banach algebra.

L1L4

2.1

If $\det (λ I - A) \neq 0$ then $B : = λ I - A$ has the two-sided inverse $\det (B)^{- 1} a d j (B)$ by [L2], so $λ \notin σ (A)$ by [L1].

step 1.1L2L1

2.2

Conversely, if $λ I - A$ has a two-sided inverse $C$ in the algebra of [step 1.1], then [L3] gives $\det (λ I - A) \det (C) = \det (I) = 1$ , so $\det (λ I - A) \neq 0$ .

step 1.1L3algebra

3.1

Combining [step 2.1] and [step 2.2] with the characterization of the spectrum in [L1]: $λ \in σ (A)$ precisely when $λ I - A$ is not invertible, which by the two steps happens precisely when $\det (λ I - A) = 0$ .

step 2.1step 2.2L1∎

Depends on

Used by

Norm need not equal spectral radius Counterexample
Riesz projection for a matrix with separated spectrum Example

Dependency tree · two levels

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Sources

Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.1.1 and §5.2.1 examples, printed pp. 209–214 and 219–222 (standard reference, not scraped)