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A regular matrix $A$ is described as a square matrix that for all positive integer $n$, is such that $A^n$ has positive entries.

How then would I prove something is regular? I mean I can prove something is irregular if $A^2$ has some 0 or negative entries; but I cant prove regularity since I cant solve $A^n$ for all integers $n$.

My thoughts are that if a matrix $A$ is diagonalisable as $A=PD^{-1}P$ then it is 'regular,' since then all $A^k$ exist; but does this also imply all entries of $A^k$ are positive?

Any hints?

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The definition is not right. A regular matrix is a matrix for which some power of the matrix has all positive entries. –  Christopher A. Wong Oct 18 '12 at 2:04
    
@ChristopherA.Wong Definitions can't be right or wrong. –  user53153 Feb 27 '13 at 3:07

4 Answers 4

If $A$ has an entry that is $0$ or negative, then $A$ is not regular. If, on the other hand, every entry in $A$ is positive, can $A^2$ have a negative or zero entry? Can $A^3$? There’s an easy proof by induction waiting here for you to find it. Note that diagonalizability has nothing to do with the matter: if $A$ is square, $A^n$ exists for all $n\ge 0$ whether or not $A$ is diagonalizable. Diagonalizability of $A$ merely makes it easy to calculate the powers of $A$.

However, that’s not the usual definition of regular matrix. The usual definition is that a square matrix $A$ is regular if it is stochastic and there is some $n\ge 1$ such that all of the entries of $A^n$ are positive.

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Linear combinations of positive numbers (with positive coefficients) will always yield positive numbers, yes? Therefore, it should suffice to note that all entries of $A$ are positive. On the other hand, if $A$ has an entry that is nonpositive, then $A$ can't be regular (since $A=A^1$ and $1>0$), so this is a sufficient condition, too.

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There is some grave confusion. A matrix is regular precisely when it is invertible. That is an $n\times n$ matrix $A$ is regular precisely when there exists some matrix $B$ such that both $AB$ and $BA$ are the identity matrix. It is absolutely not the case that a matrix is regular if the entries in its powers are positive. For example, the $2\times 2$ matrix $A=-I_2$ is regular since you can take $B=A$ and then $BA=AB=I_2$. Moreover, the $2\times 2$ matrix with all entries equal to $1$ is not regular.

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That’s one of the possible meanings of regular matrix, but I don’t think that it’s the most common one. –  Brian M. Scott Oct 18 '12 at 2:03
    
Note that definitions may vary from source to source, and that "regular" is among the most overused terms in mathematics, so it's quite possible that the OP is quoting from a source in which regular is precisely as described by the OP. I'll further remark that I personally have more frequently encountered "nonsingular" as synonymous with "invertible"--and practically never seen "regular" used in that context. –  Cameron Buie Oct 18 '12 at 2:05
    
Indeed, invertibility has little to do with powers having positive entries, so surely regular is not being used here to mean nonsingular. –  Gerry Myerson Oct 18 '12 at 12:43

A regular matrix is the same as a nonsingular matrix. A matrix M with nonnegative entries and for which all entries of M^n are positive, for some positive integer n, is said to be "primitive" [1]. Vora's definition of a regular matrix seems to be based on the definition of a primitive matrix.

[1] Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995, p. 125.

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The meaning of *regular" depends on context. It may mean "nonsingular" in some sources, and something else in other sources. –  user53153 Feb 27 '13 at 3:07

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