I am aware of the fact that every positive definite matrix $A \in \mathbb{R}^{n\times n}$ admits a Cholesky decomposition, but I am wondering if this holds for complex positive definite matrices too.

In particular, I am interested in a matrix of the form $A = \mathbb{I}- i H$ where $H$ is a complex Hermitian matrix. According to Mathworld, a necessary and sufficient condition for $A$ to be positive definite is if the Hermitian part $A_H = \frac{A + A^\dagger}{2}$ is positive definite. Since $H^\dagger = H$, here $A_H = \mathbb{I}$; therefore $A$ is always positive definite as long as $H$ is Hermitian.

So if I have a linear system $Ax=b$, can I always use Cholesky decomposition? I tried this in a C++ library (Eigen) but the results were wrong - so I have been wondering if there are any other requirements on a complex positive definite matrix for Cholesky decomposition, or if it just a coding error.

I am inclined to use Cholesky decomposition because it offers the best numerical performance in my knowledge. I'd appreciate if anyone could suggest any better alternatives.


1 Answer 1


In linear algebra literature, when a matrix is called positive definite, it is always assumed to be Hermitian. Cholesky decomposition works for and only for Hermitian positive semidefinite matrices. You see, if a matrix $A$ has a Cholesky decomposition $LL^\ast$, then $A^\ast=(LL^\ast)^\ast=(L^\ast)^\ast L^\ast=LL^\ast=A$, i.e. $A$ is necessarily Hermitian. Also, since $LL^\ast$ is a Gram matrix, it is necessarily positive semidefinite.

Most proofs about the existence Cholesky decompositions of symmetric positive definite matrices can be modified to deal with Hermitian positive definite matrices simply by changing all occurrences of matrix transposes in the proofs to Hermitian (i.e. conjugate) transposes.

In your case, $A$ is not Hermitian. Hence it cannot possibly possess any Cholesky decomposition. Some people (such as the author of the Mathworld article mentioned in your question) insist on calling matrices with positive definite Hermitian parts "positive definite". This is confusing. The spectral properties of these matrices in general are very different from those of Hermitian positive definite matrices.


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