Eigenvalue

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

A vector $v\neq 0$ satisfying this equation is an eigenvector , and all such vectors (together with $0$) form the eigenspace for $\lambda$. Eigenvalues are precisely the roots 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 $0$.
• For the identity operator $I$, the only eigenvalue is $1$.