Cholesky decomposition for special structure matrix

Question

Suppose the matrix is of the form $\tilde{A} = aI -A^TA$, where $I$ is the identity matrix and $a>eig(A^TA)$. Assume that Cholesky decomposition is possible. Do we get a nice analytic expression or low cost computation for Cholesky decomposition of $\tilde{A}$?

Answer 1

Let $A^TA = PDP^T$ be the eigen-decomposition for $A^TA$. Notice then we can write:

\begin{eqnarray*} \tilde{A} & = & aI - A^TA \\ & = & aI - PDP^T \\ & = & aPP^T - PDP^T \\ & = & P\left (aI - D\right )P^T \\ & = & P \sqrt{aI - D}\sqrt{aI-D}P^T. \end{eqnarray*}

Let $H = \sqrt{aI-D}P^T$, therefore $\tilde{A} = H^TH$ is the Cholesky decomposition.