[Math] Cholesky decomposition of a positive semi-definite

matrix-theory

We know that a positive definite matrix has a Cholesky decomposition,but I want to know how a Cholesky decomposition can be done for positive semi-definite matrices?The following sentences come from a paper.
"There are two assumptions on the specified correlation matrix R. The first is a general assumption that R is a possible correlation matrix, i.e. that it is a symmetric positive semidefinite matrix with 1’s on the main diagonal. While implementing the algorithm there is no need to check positive semi-definiteness directly, as we do a Cholesky decomposition of the matrix R at the very start. If R is not positive semi-definite, the Cholesky decomposition will fail."
Thank you for your answer.

Best Answer

You can either:

  • use a LDL^T decomposition (see e.g. here)

  • deflate the kernel yourself before: that is, compute a basis $Q_2$ for the kernel, complete it to a square orthonormal matrix $Q=[Q_1 \, Q_2]$, and assemble $$ Q^TRQ=\begin{bmatrix}R_{11} & 0\\ 0 & 0\end{bmatrix}, $$ where $R_{11}$ is going to be nonsingular (and hence can be Cholesky-factored).

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