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Suppose we have a matrix $A$ over which an inner product is defined. Let us denote this inner product by $\langle, \rangle_A$.

Now let us suppose that this matrix $A$ is symmetric. Then there exists a matrix $Q$ such that $Q^TAQ$ is a diagonal matrix. For any two vectors $v$ and $w$, is $\langle v,w\rangle_A=\langle v,w\rangle_{Q^TAQ}$?

In other words, is inner product preserved when we change the basis of $R^n$?

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    $\begingroup$ The basis has to be orthogonal. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Feb 26 at 20:33
  • $\begingroup$ @GNUSupporter8964民主女神地下教會- That's not necessarily true when diagonalizing a symmetric matrix though, right? $\endgroup$ – Anju George Feb 26 at 20:35
  • $\begingroup$ It's a well known result in linear algebra that symmetric matrices iff orthogonal eigenbasis. I'm simply addressing the last prompt alone without previous context. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Feb 26 at 20:39
  • $\begingroup$ Is the inner product defined over $A$, or the inner product defined using $A$ over $\mathbb R^n$ using a canonical basis? $\endgroup$ – Sandesh Jr Mar 1 at 10:32

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