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If we are in normed spaces say $\mathbb{R}^n$, $\langle y,x\rangle = y^Tx$ where $y^T$ is transpose of $y$. But if $A$ is an $n \times n$ matrix, then why does $\langle Ax,Ax\rangle = x^TAx$? Shouldnt it be $\langle Ax,Ax\rangle = x^TA^TAx$?

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Yes it should. $x^T A x$ does even make sense, when $A$ is $m\times n$ and $m\neq n$. – Stefan Hansen Feb 12 '13 at 7:12
Sorry A is n by n – confused Feb 12 '13 at 7:13
Don't forget to accept answers if you're satisfied with the answer given (this goes for your previous questions too). – Stefan Hansen Feb 12 '13 at 7:17
up vote 2 down vote accepted

Of course it should be $x^T A^T A x$. The only case where it would be $x^T A x$ would be if $A^T A = A$ (in particular if $A$ is an orthogonal projection).

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Hrm I guess this textbook has typo. Thanks! – confused Feb 12 '13 at 7:15
Or perhaps, depending on the context, it should have been $\langle A^Tx,x\rangle = \langle x,Ax\rangle=x^{T}Ax$. – Henning Makholm Feb 12 '13 at 7:18

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