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I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was discovered earlier, e.g. by Minkowski in 1900, so am looking for one of those original proofs.

Can you help me find this one? Is there any free eBook or skript? Also, I am not looking for the proof by Levy who proofed this for real matrices only, but another proof e.g. by Desplanques would be nice to see as well.

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  • $\begingroup$ Regular means non-singular? $\endgroup$ – Git Gud Jan 28 '14 at 1:39
  • $\begingroup$ @GitGud yes, it does $\endgroup$ – user127.0.0.1 Jan 28 '14 at 1:40
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I'm sorry that this is not the complete answer, but I found in Matrix Iterative Analysis by Varga this (see item 1.5 in Bibliography and Discussion section of Chapter 1):

That strictly or irreducibly diagonally dominant matrices are necessarily nonsingular is a result which many have discovered independently. Taussky (1949) has collected a long list of contributors to this result and is an excellent reference for related material.

Reference:

O. Taussky, A recurring theorem on determinants. Amer. Math. Monthly 56, 672--676, 1949.

I don't have access there but it seems that the paper is accessible via JStor.

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The proof by Minkowski can be found in this paper

BibTeX:

@article{Minkowski1900,
author = {Minkowski, Hermann},
journal = {Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse},
keywords = {Minkowski's determinant theorem; Minkowski unit},
language = {ger},
pages = {90-93},
title = {Zur Theorie der Einheiten in den algebraischen Zahlkörpern},
url = {http://eudml.org/doc/58467},
volume = {1900},
year = {1900},
}
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