# Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition

Let $$\mathbf{C}$$ be a $$n \times n$$ covariance matrix and assume that the LDL' Cholesky decomposition can be obtained efficiently. Can we take advantage of this to obtain a fast eigenvalue decomposition of $$\mathbf{C}$$?

As an example, $$\mathbf{C}$$ is the covariance of a discrete-time Gaussian Process (GP) having a state-space representation or that of a partially observed continuous-time state-space GP. Using Kalman filtering, the Cholesky decomposition can then be obtained with $$O(n)$$ operations instead of $$O(n^3)$$ for the general case. The eigenvalue decomposition is required to investigate the Karuhnen-Loève expansion of the GP.

To be more formal, the decomposition is $$\mathbf{C} = \mathbf{L}\mathbf{D}\mathbf{L}^\top$$ where $$\mathbf{L}$$ is lower triangular with ones on its diagonal and $$\mathbf{D}$$ is diagonal say $$\text{diag}(\mathbf{d})$$. We can assume that we have a function or a routine which for a given vector $$\mathbf{y}$$ with length $$n$$ computes efficiently the "innovation" vector $$\mathbf{e} := \mathbf{L}^{-1}\mathbf{y}$$ along with its variance vector $$\mathbf{d}$$. These two elements allow the evaluation of the likelihood for the GP example. Using this function we want to find the eigenvectors and eigenvalues.

• Fwiw, scholar.google Cholesky eigenvalue -> a paper "Mathias, Fast accurate eigenvalue computations using the Cholesky factorization, 1996"; I don't understand the conditions offhand. – denis Jan 29 '20 at 11:32
• Thanks @denis. This is a step forward and I should read this. – Yves Jan 29 '20 at 14:13