# 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.

• THANK YOU .but here I want to know if a positive semi-deﬁnite can be done for Cholesky decomposition? and how? Jan 20, 2014 at 10:39
• "that how a positive semi-deﬁnite be done for Cholesky decomposition"??? What do you mean? Jan 20, 2014 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." Jan 20, 2014 at 11:46
• Do you mean to ask why R has a Cholesky decomposition if and only if R is positive semidefinite? Jan 21, 2014 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. Jan 22, 2014 at 20:08

• 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).
$$P^T R P = R_1^T R, \quad R_1 = \begin{bmatrix} R_{11} & R_{22} \\ 0 & 0 \end{bmatrix}.$$