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I would like to ask a question about singular values of matrices of the form $A^TA$. We know that by Courant minimax principle the singular values are given by (in increasing order $s_1 > s_2...>s_n$

$(s_i)^2 = min_{\{V | dim(V) \geq n-i+1\}}max_{\{x | |x|=1, x \in V\}}\{|Ax|\}$

Also by Cauchy interlacing property if we have an operator like B = A.P (were P is a projection) then we can relate the singular value of B to that of A since

$B^TB = PA^TAP$

so that singular values of B and A may be compared through Cauchy interlacing property. However what I have information about is the operator B=P.A (where P is a projection) and the singular values of this operator are related to eigenvalues of $A^T.P.P.A = A.P.A$. Is there anyway to relate the eigenvalues of $A^T.P.A$ to $A^TA$.

i.) I know that there are bounds for singular values of product of matrices of the form for instance given in

Relation between singular values of matrices and their products

but they do not turn out to be good for me.

ii.) Trying $B.B^T$ does not also help since then I would need information about $A^T.P$ which I dont have.

I would be grateful if you could direct me at such theorems as Cauchy interlacing property that might be related to this, if there are any ofcourse. Thanks

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