# Cholesky decomposition of a positive semi-deﬁnite

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.

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THANK YOU .but here I want to know if a positive semi-deﬁnite can be done for Cholesky decomposition? and how? –  Purple Jan 20 '14 at 10:39
"that how a positive semi-deﬁnite be done for Cholesky decomposition"??? What do you mean? –  Dima Pasechnik Jan 20 '14 at 11:28
excuse for my bad english."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." –  Purple Jan 20 '14 at 11:46
Do you mean to ask why R has a Cholesky decomposition if and only if R is positive semidefinite? –  Dima Pasechnik Jan 21 '14 at 15:55
en.m.wikipedia.org/wiki/Cholesky_decomposition has a proof that yes, indeed, it is correct, a p.s.d. R will have a Cholesky decomposition. –  Dima Pasechnik Jan 22 '14 at 20:08

## 1 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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