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What proprieties does a square $n\times n$ real matrix $\mathbf M$ need to have in order to have all it's eigenvalues be less than or equal to one in absolute value?

I'm looking for proprieties such as "Have all it's elements be less than one" or "The sum of the squares of it's columns have to add up to one or less" or similar proprieties.

[Note: I am not claiming these examples to be true, I am using them merely as demonstrations of the kind proprieties I am looking for]

Much appreciated.

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  • $\begingroup$ Are you looking for necessary conditions or sufficient conditions? $\endgroup$ – Ian Coley Mar 23 '14 at 18:54
  • $\begingroup$ Any conditions would help, really. $\endgroup$ – Disousa Mar 23 '14 at 18:55
  • $\begingroup$ Are you talking about real or complex matrices? For real matrices, the condition is much weaker, because it doesn't have to have eigenvalues at all (if $n$ is even). $\endgroup$ – fgp Mar 23 '14 at 18:55
  • $\begingroup$ Real Matrices. I'll edit the question to reflect this. $\endgroup$ – Disousa Mar 23 '14 at 18:56
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    $\begingroup$ Have you heard of the Gershgorin's circle theorem? That should answer your question. $\endgroup$ – Samrat Mukhopadhyay Mar 23 '14 at 19:05
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Instead of summing the squares of elements in a column or row, sum the absolute values of the elements in a row. if this is less than 1 for each row, you have it. Same for columns. these correspond to induced norms,

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  • $\begingroup$ So ANY matrix that respects this will have $\lambda_i\leq 1$? $\endgroup$ – Disousa Mar 23 '14 at 21:15
  • $\begingroup$ well, $|\lambda_i|$ in case they are complex. But yes. $\endgroup$ – Will Jagy Mar 23 '14 at 21:17

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