$A$ is a symmetric matrix, $x$ is a vector. If $(Ax,Ax)=(x,x)$,then $Ax=x$ or $Ax=-x$. Is it true? How to prove it?

Question

$A$ is a symmetric matrix, $x$ is a vector. If $(Ax,Ax)=(x,x)$,then $Ax=x$ or $Ax=-x$.Is it true? How to prove it? or give some examples. Thanks!

What are the parentheses here? Are you perhaps meaning to use an inner product? More commonly notated as $\langle Ax,Ax\rangle$? — JMoravitz, Dec 06 '19 at 14:47

Arthur · Accepted Answer · 2019-12-06T15:01:14.003

1

This is not true. Consider the matrix $$ A = \begin{bmatrix}1&0\\0&-1\end{bmatrix} $$ We do have $(Ax, Ax) = (x, x)$ for any $x\in \Bbb R^2$, but a vector like $\left[\begin{smallmatrix}1\\1\end{smallmatrix}\right]$ doesn't get sent to itself or its negative.

edited Dec 06 '19 at 15:01

answered Dec 06 '19 at 14:55

Arthur

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$A$ is a symmetric matrix, $x$ is a vector. If $(Ax,Ax)=(x,x)$,then $Ax=x$ or $Ax=-x$. Is it true? How to prove it?

1 Answers1