Suppose it is known that a matrix $B$ of size $(ns)\times(ns)$ has the form $B=\begin{bmatrix} A_1 \\ \vdots \\ A_n \end{bmatrix}\begin{bmatrix} A_1^* & \cdots & A_n^* \end{bmatrix}$, where $A_1, \cdots, A_n$ are block matrices of size $s \times s$. How is it possible to determine these $A_i$'s?
It is stated in the book Positive Trigonometric Polynomials and Signal Processing Applications, 2nd Edition, by Bogdan Dumitrescu on page 270 that this can be done by eigenvalue decomposition or Cholesky factorization with pivoting. How to do this explicitly is not evident to me. I would be very grateful for assistance in understanding this. Thank you.
Thank you.
$\DeclareMathOperator{\diag}{diag}$ More generally, suppose it is known that $B$ is $N$-by-$N$ and positive semi-definite of rank $\le s$ (where $N$ is not necessarily a multiple of $s$.) Then there exists a unitary $N$-by-$N$ matrix $U$ such that $U^* B U = \diag(d_1, \dots, d_s, 0, 0, \dots)$, with $d_i \ge 0$. So $$ B = U \diag(\sqrt{d_1}, \dots, \sqrt{d_s}, 0, \dots, 0)^2 U^*. $$ Take $A$ to be the first $s$ columns of $U \diag(\sqrt{d_1}, \dots, \sqrt{d_s}, 0, \dots, 0)$. Then $B = A A^*$.
Finally, the computation of $U$ is equivalent to computing an orthonormal basis of eigenvectors of $B$.
$A$ is not unique as it could be replaced by $A V$ where $V$ is $s$-by- $s$ unitary.