A real symmetric positive semi-definite matrix $A$ can be decomposed in the form

$A = P^TLDL^TP$,

where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix.

**Questions**

**(a)** Is this decomposition unique?

**(b)** Let $B = \Pi^TA\Pi$, where $\Pi$ is a permutation matrix. It follows that

$B = \Pi^TP^TLDL^TP\Pi = Q^TLDL^TQ$,

where $Q = \Pi P$ is a permutation matrix, too. If the LDLT decomposition is unique, this means that it is invariant to simultaneous permutations of rows and columns of a matrix, right?