If $A$ and $B$ are symmetric matrices related by $A = P^tBP$, where P is invertible, does it follow that the ranks of $A$ and $B$ are equal?
equal ranks for symmetric matrices of the form $A = P^tBP$?
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linear-algebra
matrices
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0In fact multiplying by an invertible matrix does not change rank, hence $\operatorname{rank}(B)=\operatorname{rank}(BP)=\operatorname{rank}(P^TBP)$. – 2012-09-11
1 Answers
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Hint: $x\in \operatorname{Ker}A$ then $Px\in \operatorname{Ker}B$. $x\in \operatorname{Ker}B$ then $P^{-1}x\in \operatorname{Ker}A$. Conclude that $\operatorname{dim} (\operatorname{Ker}A)= \operatorname{dim}(\operatorname{Ker}B)$