# Gaussian elimination for homogeneous linear systems

• I
it is quite rare that ##Q## and ##P## commute
Sorry you are right, it was my fault

Trying again: the permutation matrix ##P## needed to reorder ##A## eigenvalues has the special property ##P=P^T=P^{-1}##.

##E=PE_1P##, ##E_1## is the ##A## eigenvalues diagonal matrix ##E## reordered with its ##k## possibly zero eigenvalues as last ##k## entries.

it turn out the following:
##A=QEQ^T##
##A=QPE_1PQ^T##
##PAP^T=PQPE_1PQ^TP^T=PQPE_1P^TQ^TP^T=(PQP)E_1(PQP)^T##

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Sorry you are right, it was my fault

Trying again: the permutation matrix ##P## needed to reorder ##A## eigenvalues has the special property ##P=P^T=P^{-1}##.

##E=PE_1P##, ##E_1## is the ##A## eigenvalues diagonal matrix ##E## reordered with its ##k## possibly zero eigenvalues as last ##k## entries.

it turn out the following:
##A=QEQ^T##
##A=QPE_1PQ^T##
##PAP^T=PQPE_1PQ^TP^T=PQPE_1P^TQ^TP^T=(PQP)E_1(PQP)^T##
Maybe that result really does not help much.
I found the following https://mathoverflow.net/questions/155147/cholesky-decomposition-of-a-positive-semi-deﬁnite
Neverthless it seems to apply to just symmetric positive semi-definite matrices (p.s.d), right ?