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I was studying matrices and I came to know the following two results:

  1. Real symmetric matrices, $A = P D P^{-1} = P D P^T$ where $D$ is diagonal
  2. For any real matrix $A$ and any vectors $x$ and $y$, $\langle A\mathbf{x},\mathbf{y}\rangle = \langle\mathbf{x},A^T\mathbf{y}\rangle.$

Please explain me the meaning of second and proof of the first. In general I want to understand the meaning of answer given here.

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Vedanshu
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  • What do you mean by meaning of the second? It means what it's written: inner product of $Ax$ and $y$ is exactly inner product of $x$ and $A^T y$. – Balarka Sen Dec 19 '16 at 10:19
  • By its meaning I meant, what does the inner product of $Ax$ and $y$ signify ? Also its proof. – Vedanshu Dec 19 '16 at 10:22
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    I don't know how to interpret it in some significant way. It's just the definition of the standard inner product on $\Bbb R^n$: $(Ax) \cdot y = (Ax)^T y = x^T A^T y = x \cdot (A^T y)$. – Balarka Sen Dec 19 '16 at 10:43
  • What does $\langle A\mathbf{x},\mathbf{y}\rangle$ signifies ? – Vedanshu Dec 28 '16 at 13:44

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