Linear and multilinear algebra; matrix theory Definition

Eigenvalue

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $T \colon V \to V$ be an endomorphism. A scalar $\lambda \in F$ is an eigenvalue of $T$ when there is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v.$ An eigenvalue of an $n \times n$ matrix $A$ over $F$ is an eigenvalue of the map $F^n \to F^n$ , $x \mapsto Ax$ .

Remarks

So $\lambda$ is an eigenvalue of $T$ if and only if $T - \lambda \id_V$ is not injective. In finite dimensions, this holds if and only if $\det(T - \lambda \id_V) = 0$ . The eigenvalues of the diagonal matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ are $2$ and $3$ . The rotation of $\R^2$ by $90$ degrees has no real eigenvalue.

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Vector space definition
Linear map definition
Endomorphism definition
Eigenvalue definition

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43 pages build on this one.

Tag	`0ZND7NQ970`
URI	`m2://0ZND7NQ970`
Kind	definition
Writing	W0 stub
MSC	15A18
Defines	Eigenvalue
Uses	Endomorphism

Eigenvalue

Definition

Remarks

Reading path

Where this leads

Further reading

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