A is n×n symmetric matrix.
its all eigen values are greater than 0.
Characteristics polynomial is det(A−λIn)=0
λ are the eigenvalues of A.
As P is orthogonal matrix so PT=P−1
Characteristics polynomial for PTAP is given as-
det(PTAP−λIn)=det(P−1AP−λP−1P)=detP−1(A−λIn)P=det(P−1)det(P)det(A−λIn)=det(P−1P)det(A−λIn)=det(A−λIn)
(PTAP)T=(PA)T(PT)T=PTATP=PTAP,asAT=A
So PTAP is symmetric matrix.
From characteristics equation, eigenvalues of PTAP is the same as eigenvalues of A.
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