I wish to express, $x^TAx$ ($x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n\times n}$) which is a scalar quantity as product of two vectors, i.e, $v^Ty$, where $v$ should only contain elements of $x$ and $y$ should only contain elements of $A$. How can I accomplish that? Is it possible to write this scalar quantity as product of two vectors?
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$\begingroup$ How about $v=x$ and $y=Ax$? $\endgroup$– DavidAug 20, 2019 at 5:07
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$\begingroup$ No! that wont do, I should have made it clear, y must contain only A and not both A and x $\endgroup$– dead_spaceAug 20, 2019 at 5:08
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$\begingroup$ @Amardeepmishra, what does it mean "$y$ must contain only $A$"? $A$ is a matrix, $y$ is a vector as well as $Ax$. $\endgroup$– HasekAug 20, 2019 at 5:11
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$\begingroup$ @Hasek, $y$ should contain only the elements of $A$, i.e either individual elements, rows or columns. $\endgroup$– dead_spaceAug 20, 2019 at 5:15
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1 Answer
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You can do it in ${\Bbb R}^{n^2}$ with $$v=(x_1^2,x_1x_2,x_1x_3,\ldots,x_1x_n,x_2x_1,x_2^2,x_2x_3,\ldots,x_2x_n,\ldots,x_n^2)$$ and $$y=(a_{11},a_{12},a_{13},\ldots,a_{1n},a_{21},a_{22},a_{23},\ldots,a_{2n},\ldots,a_{nn})\ .$$
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