1
$\begingroup$

I read from somewhere that the quadratic form of a matrix $A$, which is the $\vec{x}^\top A \vec{x}$ can be bounded by their eigenvalues:

$$\lambda_{\min} \| \vec{x} \|_2^2 \leq \vec{x}^\top A \vec{x} \leq \lambda_{\max} \| \vec{x} \|_2^2$$

where $\lambda_{\min}$ and $\lambda_{\max}$ are the minimum and maximum eigenvalues of $A$. I saw its proof from somewhere, but I was hoping for a more reliable source, like a book, a paper or a textbook that I can formally and easily cite (as opposed to lecture notes or slides) with BibTeX. Also, does this theorem have a name? I don't know what to call it. Thanks!

$\endgroup$

1 Answer 1

1
$\begingroup$

I assume that $A$ is a real symmetric matrix, correct? Otherwise, the eigenvalues might not be real.

If you are looking for a textbook, see Olver-Shakiban: Applied Linear Algebra (2nd edition), Chapter 8, Theorem 8.40.

https://link.springer.com/book/10.1007/978-3-319-91041-3

I don't know if the result has a standard name. In that book, it's called optimization principle for eigenvalues.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .