# Remind me of the name of this matrix inequality

**URL:** <https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500>\
**Category:** Factual Questions\
**Created:** [May 6, 2009, 1:49am UTC](https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500 "2009-05-06T01:49:01Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [May 6, 2009, 1:49am UTC](https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500/1 "2009-05-06T01:49:01Z")

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For any matrix A and vector x, the quadratic form x[sup]T[/sup]Ax is bounded above by the product of the maximum magnitude of any eigenvalue of A and the norm of x, and below by the product of the minimum magnitude of any eigenvalue of A and the norm of x. I can’t for the life of me remember what this is called.

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**Author:** ![Arjuna34](https://avatars.discourse-cdn.com/v4/letter/a/eb8c5e/32.png) [@Arjuna34](https://boards.straightdope.com/u/Arjuna34)\
**Post date:** [May 6, 2009, 4:45am UTC](https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500/2 "2009-05-06T04:45:36Z")

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Are you sure that inequality has a name? It’s not named here:[Modelling and control of robot manipulators](http://books.google.com/books?id=v9PLbcYd9aUC&pg=PA346&lpg=PA346&dq=%22quadratic+form%22+matrix+inequality+eigenvalue+-riccati&source=bl&ots=tvn6LSLmOG&sig=FgAUZZi3MkbDgZ3yMugPnnGGSMU&hl=en&ei=ABABStrJGM6fmAfh1YySCA&sa=X&oi=book_result&ct=result&resnum=5#PPA346,M1).

Is it a special case of the Wielandt inequality [(which is supposedly equivalent to the Kantorovich inequality)](http://www.informaworld.com/smpp/content~content=a778396027~db=all)?

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**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\
**Post date:** [May 6, 2009, 5:06am UTC](https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500/3 "2009-05-06T05:06:24Z")

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Are you thinking of the [Hausdorff-Toeplitz theorem](http://en.wikipedia.org/wiki/Numerical_range)?

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [May 6, 2009, 12:10pm UTC](https://boards.straightdope.com/t/remind-me-of-the-name-of-this-matrix-inequality/495500/4 "2009-05-06T12:10:32Z")

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> [@Omphaloskeptic](#):
>
> Are you thinking of the [Hausdorff-Toeplitz theorem](http://en.wikipedia.org/wiki/Numerical_range)?

It’s related to that, but not exactly the same. But that page did jog my memory: it’s the [Rayleigh-Ritz](http://www.physics.arizona.edu/~restrepo/475A/Notes/sourcea/node79.html) inequality (and only holds for Hermitian matrices).