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.

Explanations

L1 human

Most vectors change direction when a linear map acts on them. An eigenvalue is a scaling factor $\lambda$ such that some nonzero vector only gets stretched by $\lambda$, without a change of its line. These special directions often reveal the structure of the map.