# Algorithms for Nonunique Factorizations of a Real Symmetric Matrix

Suppose we have an indefinite real symmetric matrix $P$, of (reduced) rank $\alpha$. Then there is a (nonunique) decomposition of the matrix:

$P = YMY'$, where $M = \left[\begin{array}{cc} M_+ & 0 \\ 0 & M_-\end{array}\right]$, where $M_+$ is positive definite and $M_-$ is negative definite.

Rank $M$ = $\alpha$.

I am reading an old control paper: Morf et al. (1974) "Some New Algorithms for Recursive Estimation in Constant, Linear, Discrete-Time Systems," which uses this decomposition, but does not say how to operationalize it. I assume that I am completely missing something, because I have no idea how to implement it.

I have thought about picking sizes for $M_+$ and $M_-$, say of rank $\alpha/2$ each. Then we have

$P = Y_1M_{+}Y_1' + Y_2M_-Y_2'$

where $Y = [Y_1;Y_2]$ is a conforming partition for $Y$...but I cannot think of anywhere to go from here.

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Actually, you can demand that your $M$ be diagonal, the entries being the eigenvalues. Since your $P$ is symmetric and real, there is an orthogonal basis of eigenvectors, even when eigenvalues have multiplicity other than one. I should point out that your conditions allow for the possibility of $0$ as an eigenvalue, in which case the diagonal actually has some zeroes. So I am guessing they have something faster in mind, see SYLVESTER and CHOLESKY