Testwiki:Reference desk/Archives/Mathematics/2024 September 19

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September 19

Property of a matrix

If I raise a permutation matrix to a high power it is still a permutation matrix (all values 0 or 1). If I raise a stochastic matrix to a high power it is still a stochastic matrix (all values <=1). But there are a lot of matrices raised to a high power have elements with very large values. Is there a name for the property of a matrix that when raised to a high power remain reasonably valued? RJFJR (talk) 22:12, 19 September 2024 (UTC)

The spectrum of such a matrix is inside the closed unit disk (a necessary but not sufficient condition). A matrix whose spectrum is in the open unit disk always satisfies this property. I'm not sure there is a general name for this property though. Tito Omburo (talk) 23:07, 19 September 2024 (UTC)

That's it. Thank you! RJFJR (talk) 02:08, 20 September 2024 (UTC)

Note that when the spectral radius is strictly less than $1,$ the powers of the matrix tend to the null matrix, which is not very exciting. I also vaguely remember seeing a theorem that states or implies that when the spectral radius is equal to $1,$ the growth of the entries is polynomial in the value of the exponent.  --Lambiam 21:55, 20 September 2024 (UTC)