Definition · applies to square matrices and linear maps

High school

Consider a square matrix $A$, which stretches and turns vectors when it multiplies them. An eigenvalue of $A$ is a number $\lambda$ such that $A$ stretches some nonzero vector $v$ by the factor $\lambda$ without turning it off its line:

\[A v = \lambda v.\]
Caveat. An eigenvalue can be a complex number, with a complex eigenvector. Then the picture of stretching without turning does not apply in the real plane: for example, the matrix that rotates the plane by $90°$ turns every real vector off its line, but it has the complex eigenvalues $i$ and $-i$.
Undergraduate

Consider a square matrix $A$. An eigenvalue of $A$ is a number $\lambda$ such that

\[A v = \lambda v\]

for some nonzero vector $v$, called an eigenvector.

Examples

Worked examples

  • The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) : $i$ and $-i$ High school
    The eigenvalues of \(A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) are $i$ and $-i$, because \(A \begin{bmatrix} 1 \\ -i \end{bmatrix} = \begin{bmatrix} i \\ 1 \end{bmatrix} = i \begin{bmatrix} 1 \\ -i \end{bmatrix}\) and \(A \begin{bmatrix} 1 \\ i \end{bmatrix} = \begin{bmatrix} -i \\ 1 \end{bmatrix} = -i \begin{bmatrix} 1 \\ i \end{bmatrix}\). $A$ rotates every nonzero real vector by $90°$, so it has no real eigenvalues.
  • The matrix \(\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}\) : None Undergraduate
    The matrix \(A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}\) has no eigenvalues, because only square matrices have eigenvalues.
  • The matrix \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\) : $2$ and $-3$ Undergraduate
    The eigenvalues are $2$ and $-3$, because \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -3 \end{bmatrix} = -3 \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).
  • The matrix \(\begin{bmatrix} 3 & 1 \\ -1 & 1 \end{bmatrix}\) : $2$ (twice) Undergraduate
    The eigenvalue of \(A = \begin{bmatrix} 3 & 1 \\ -1 & 1 \end{bmatrix}\) is $2$, because \(A \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ -2 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ -1 \end{bmatrix}\).
  • The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) : $1$ (twice) Undergraduate
    The shear \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) has the eigenvalue $1$ twice, and it fixes the vector $(1, 0)$.

Properties

  • The eigenvalues of $A$ are the roots of its characteristic polynomial $\det(\lambda I - A)$.
  • The origin of $\dot{x} = A x$ is asymptotically stable if and only if every eigenvalue of $A$ has negative real part ($A$ is Hurwitz).

Distinctions involving this term

  • Eigenvalue vs. singular value: Eigenvalues and singular values both describe how a matrix stretches vectors, but they agree only for special matrices. An eigenvalue of a square matrix $A$ is a factor $\lambda$ by which $A$ scales a particular nonzero vector $x$, called an eigenvector: $Ax = \lambda x$. A singular value of a matrix $A$, square or not, is a factor $\sigma$ by which $A$ stretches vectors along some direction, allowing them to turn. Singular values are real and nonnegative, but eigenvalues can be negative or complex.

All definitions · All distinctions · All examples