Eigenvalue - Knowlpedia

An eigenvalue of a linear operator $T:V\to V$ is a scalar $\lambda\in\mathbb{F}$ such that there exists a nonzero vector $v\in V$ with

T(v)=\lambda v.

A vector $v\neq 0$ satisfying this equation is an eigenvector, and all such vectors (together with $0$ ) form the eigenspace for $\lambda$ . If $V$ is finite-dimensional, the eigenvalues in $\mathbb F$ are precisely the roots in $\mathbb F$ of the characteristic polynomial.

Examples

For a diagonal matrix $\operatorname{diag}(d_1,\dots,d_n)$ , the eigenvalues are $d_1,\dots,d_n$ .
For the projection $P(x,y)=(x,0)$ on $\mathbb{R}^2$ , the eigenvalues are $1$ and $0$ .
On a nonzero vector space, the identity operator $I$ has $1$ as its only eigenvalue.