Definition · applies to symmetric positive definite matrices · also called Cholesky factorization

Undergraduate

Consider a symmetric matrix $A$ whose eigenvalues are all positive (that is, $A$ is positive definite). The Cholesky decomposition of $A$ is the factorization $A = L L^{\top}$, where $L$ is lower triangular with positive diagonal entries.

Examples

Worked examples

  • The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) : None, since $A$ is not symmetric Undergraduate
    $A$ has an LU decomposition, because subtracting $2$ times row $1$ from row $2$ makes it upper triangular, and $L$ records the $2$. $A$ has the distinct eigenvalues $\frac{5 \pm \sqrt{17}}{2}$, so it is diagonalizable, and its Jordan–Chevalley decomposition has $D = A$ and $N = 0$.
  • The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) : \[L = \begin{bmatrix} 2 & 0 \\ 1 & \sqrt{2} \end{bmatrix}\] Undergraduate
    The matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is symmetric with positive eigenvalues $\frac{7 \pm \sqrt{17}}{2}$, so it has a Cholesky decomposition. Its factor $L$ is the LU factor $U$ with each row divided by the square root of its diagonal entry, transposed.
  • The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) : None, since $A$ is not symmetric Undergraduate
    $A$ is upper triangular, so its LU decomposition is $I \cdot A$, and its QR decomposition is $I \cdot A$. $A$ is not diagonalizable, so its Jordan–Chevalley decomposition has a nonzero nilpotent part $N$.
  • The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) : None, since $A$ has the negative eigenvalue $-1$ Undergraduate
    The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) is symmetric but has no Cholesky decomposition, because its eigenvalue $-1$ is negative, so it is not positive definite.

Properties

  • A real square matrix $A$ has a Cholesky decomposition exactly when $A$ is symmetric positive definite, and then the decomposition is unique.
  • Computing a Cholesky decomposition takes about half the work of an LU decomposition, and it needs no row exchanges.
  • If $z$ is a vector of independent standard normal random variables and $\Sigma = L L^{\top}$, then $\mu + L z$ is normally distributed with mean $\mu$ and covariance $\Sigma$.

Notes

The related $L D L^{\top}$ decomposition, with $L$ lower triangular with ones on its diagonal and $D$ diagonal, avoids square roots and also applies to some symmetric matrices that are not positive definite.

For complex matrices, $A$ is Hermitian positive definite and $A = L L^*$.

Distinctions involving this term

  • Types of matrix decompositions: A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.

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