Matrices can rotate, reflect, shear, stretch, and squash. Amid all that motion, some directions are special: the matrix stretches or flips vectors on those lines but does not knock them onto a different line. Those directions are spanned by eigenvectors; the stretch factors are eigenvalues.
The defining equation
A nonzero vector $\mathbf{v}$ is an eigenvector of square matrix $A$ with eigenvalue $\lambda$ when $A\mathbf{v}=\lambda\mathbf{v}$. Applying $A$ has the same effect as simply multiplying by the scalar $\lambda$. Rearranged, $(A-\lambda I)\mathbf{v}=\mathbf{0}$, so $\lambda$ is an eigenvalue precisely when $A-\lambda I$ fails to be invertible — that is, when $\det(A-\lambda I)=0$.
Example Stretch the plane by $2$ in the $x$-direction and $3$ in the $y$-direction: $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$. Then $\mathbf{e}_1=(1,0)$ is an eigenvector with $\lambda=2$, and $\mathbf{e}_2=(0,1)$ with $\lambda=3$. Most other vectors get both stretched and realigned relative to the axes — their direction changes, so they are not eigenvectors.
Why anyone cares
- Diagonalization. If you have a basis of eigenvectors, $A$ becomes a diagonal matrix in those coordinates — powers and exponentials of $A$ become easy.
- Dynamics. For discrete systems $\mathbf{x}_{n+1}=A\mathbf{x}_n$, eigenvalues decide growth, decay, and oscillation.
- Stability. In continuous linear systems $\mathbf{x}'=A\mathbf{x}$, signs of real parts of eigenvalues govern whether solutions blow up or settle.
Build the surrounding vocabulary in Linear algebra and follow the first-course pathway.
Citations & further reading
- Strang, G. Introduction to Linear Algebra. Wellesley-Cambridge. Eigenvalue chapters with strong geometry. math.mit.edu/~gs/linearalgebra
- 3Blue1Brown, Essence of linear algebra — eigenvalues episodes. YouTube playlist
- MIT OCW 18.06 Linear Algebra. ocw.mit.edu
- Khan Academy, Eigenvalues and eigenvectors unit. khanacademy.org