Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A$ be a real symmetric matrix and $\lambda$ be its eigenvalue.

For a unit vector $x$, if $x^TAx=\lambda$, is it true that $Ax=\lambda x$?

share|cite|improve this question

1 Answer 1

up vote 5 down vote accepted

No. Let $A$ be the diagonal matrix with entries $2,1,0$. Let $X = \begin{pmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2}\end{pmatrix}$. Clearly $X$ is a unit vector.

$1$ is an eigenvalue of $A$. $X^TAX = 1$. But $AX = \begin{pmatrix} \sqrt{2} \\ 0 \\ 0\end{pmatrix}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.