Definition · applies to matrices

Undergraduate

Consider an $m \times n$ real matrix $A$. The singular values of $A$ are the square roots of the eigenvalues of the $n \times n$ matrix $A^{\top} A$, which are nonnegative.

Examples

Worked examples

  • The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) : $1$ and $1$ High school
    The singular values of \(A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) are $1$ and $1$, because $A^{\top} A = I$, whose eigenvalues are $1$ and $1$. A rotation does not change the length of any vector.
  • The matrix \(\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}\) : $\sqrt{14}$ and $1$ Undergraduate
    The singular values of \(A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}\) are $\sqrt{14}$ and $1$, the square roots of the eigenvalues of \(A A^{\top} = \begin{bmatrix} 5 & 6 \\ 6 & 10 \end{bmatrix}\). That matrix has trace $15$ and determinant $50 - 36 = 14$, so its eigenvalues are the roots of $\lambda^2 - 15 \lambda + 14 = (\lambda - 14)(\lambda - 1)$. The $3 \times 3$ matrix $A^{\top} A$ has the same nonzero eigenvalues, plus $0$.
  • The matrix \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\) : $3$ and $2$ Undergraduate
    The singular values of \(A = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\) are $3$ and $2$, the square roots of the eigenvalues $9$ and $4$ of \(A^{\top} A = \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix}\). $A$ is symmetric, so its singular values are the absolute values of its eigenvalues.
  • The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) : $\varphi$ and $1 / \varphi$, where $\varphi = \frac{1 + \sqrt{5}}{2}$ Undergraduate
    The shear \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) has the eigenvalue $1$ twice, and it fixes the vector $(1, 0)$. Its singular values are $\varphi \approx 1.618$ and $1 / \varphi \approx 0.618$, the square roots of the eigenvalues $\frac{3 \pm \sqrt{5}}{2}$ of $A^{\top} A$, because the shear stretches some directions and shrinks others.

Properties

  • Geometrically, $A$ maps the unit sphere to an ellipsoid, and the lengths of the ellipsoid’s semi-axes are the singular values of $A$.
  • The largest singular value of $A$ is its operator norm $\lVert A \rVert_2$, the most that $A$ stretches a vector.
  • For a symmetric matrix, the singular values are the absolute values of the eigenvalues.

Distinctions involving this term

  • Eigenvalue vs. singular value: An eigenvalue of a square matrix $A$ is a factor $\lambda$ by which $A$ scales a particular nonzero vector $x$, called a eigenvalue: $Ax = \lambda x$. A singular value of any matrix $A$ is a factor $\sigma$ by which $A$ stretches vectors along some direction, allowing it them turn. Singular values are real and nonnegative, but eigenvalues can be negative or complex.

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