Eigenvalue

Let $T$ be a linear operator $T:V\to V$ over a field $F$. A scalar $\lambda\in F$ is an eigenvalue of $T$ if there exists a nonzero vector $v\in V$ such that

Any such $v\neq 0$ is an eigenvector of $T$ with eigenvalue $\lambda$.

Eigenvalues can be detected via the characteristic polynomial (over a field where that polynomial splits). The set of eigenvalues can depend on the base field; passing from $\mathbb{R}$ to the complex numbers can create eigenvalues that were not previously available.

Examples:

• If $T=\lambda I$ (scalar multiplication) on $F^n$, then $\lambda$ is an eigenvalue and every nonzero vector is an eigenvector.
• The rotation of $\mathbb{R}^2$ by $90^\circ$ has no real eigenvalues; viewed over $\mathbb{C}^2$ it has eigenvalues $i$ and $-i$.
• The matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ (acting on $F^2$) has eigenvalue $1$.