Linear and multilinear algebra; matrix theory Definition

Spectrum of a matrix

Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$ . The spectrum of $A$ is the set $\sigma(A) = \{\lambda \in F : A - \lambda I_n \text{ is not invertible}\}.$ It is the set of eigenvalues of $A$ .

Remarks

The spectrum has at most $n$ elements, because each eigenvalue is a root of the characteristic polynomial of degree $n$ . Over $\C$ , the spectrum of each square matrix is nonempty. The spectrum depends on the field: the real matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has empty spectrum over $\R$ and spectrum $\{i, -i\}$ over $\C$ . Some authors count the eigenvalues with their multiplicities and call this list the spectrum.

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Vector space definition
Linear map definition
Endomorphism definition
Eigenvalue definition
Spectrum of a matrix definition

Tag	`04EHB7YDQS`
URI	`m2://04EHB7YDQS`
Kind	definition
Writing	W1 stated
MSC	15A18
Defines	Spectrum of a matrix
Uses	Eigenvalue

Spectrum of a matrix

Definition

Remarks

Reading path

Further reading

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