Linear and multilinear algebra; matrix theory Definition

Eigenvector

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$ . An eigenvector of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v$ . Then $\lambda$ is an eigenvalue of $T$ .

Remarks

Each nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue. For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ , the vector $(1, 1)^T$ is an eigenvector for $3$ , because $A (1, 1)^T = (3, 3)^T$ . The zero vector is never an eigenvector, by convention.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Vector space definition
Linear map definition
Endomorphism definition
Eigenvalue definition
Eigenvector definition

Where this leads

24 pages build on this one.

Tag	`0B5H4208Z9`
URI	`m2://0B5H4208Z9`
Kind	definition
Writing	W0 stub
MSC	15A18
Defines	Eigenvector
Uses	Eigenvalue

Eigenvector

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by