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I observed experimentally that, when generating (pseudo) random matrices, their condition number increases with the dimension. Why?

When using the norm induced by the Euclidean norm for vector spaces, the condition number is the absolute value of the ratio between the greatest and the smallest eigenvalue. It seems to me that, since increasing the dimension implies getting more eigenvalues, it is easier that this ratio increases.

Rodrigo de Azevedo
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iamanoob
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1 Answers1

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This is correct, and is a (very famous) result of Alan Edelman's from his PhD thesis:

Edelman, Alan, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9, No. 4, 543-560 (1988). ZBL0678.15019.

Igor Rivin
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  • Thanks! I don't have access to the paper right now. Could you confirm that not only my experimental observation is true, but my reasoning is ok? – iamanoob Oct 16 '22 at 17:45
  • @iamanoob You can easily find Edelman's PhD thesis (no paywall), and as for your reasoning, there is not enough of it to comment sensibly... – Igor Rivin Oct 16 '22 at 18:00
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    Why not link to the PhD thesis? http://hdl.handle.net/1721.1/14322 – Rodrigo de Azevedo Oct 18 '22 at 08:53
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    @RodrigodeAzevedo Because the published paper is more on point, but thank you for the link! – Igor Rivin Oct 18 '22 at 13:23
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    @IgorRivin My point was that the persistent link, even if not accompanied by a full reference, has nonzero value. From my point of view, I invested a few seconds in finding the persistent link and there is no reason for others to make that investment. – Rodrigo de Azevedo Oct 18 '22 at 14:46
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    @RodrigodeAzevedo And my view is that anyone who is too lazy to chase down a link from a reference deserves what they get. Also, in my considerable experience, no link is truly persistent. – Igor Rivin Oct 18 '22 at 16:00
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    @IgorRivin I don't see it as a matter of laziness. For example, linking to the thesis may lead search engines to list your answer with higher priority whenever someone searches for Edelman's thesis. – Rodrigo de Azevedo Oct 18 '22 at 17:12
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    Thank you both, for the link to the paper, and the link to the thesis !

    In particular, I had found a link to the thesis on Google but which led to a crappy PDF with bad OCR I was unable to find, whereas @RodrigodeAzevedo link led me to a nicer PDF that I could OCR without issues.

     – a06e Feb 01 '25 at 16:04