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Stochastic matrices $\bf P$ with element $p_{i,j}$ at row $i$ and column $j$ have some specific limitations to them, namely:

$$\cases{\displaystyle\sum_{\forall j} p_{i,j} = 1\\ p_{i_j} \geq 0}$$

Is there some way we can use these limitations to derive what restrictions will on their inverse ${\bf P}^{-1}$.

The properties I know of so far are limited to all $\lambda({\bf P}) \in [0,1]$ with one guaranteed eigenvalue always located at $1$ with corresponding eigenvector representing a steady-state. But there can exist several of these. For example the matrix $${\bf P} = \begin{bmatrix}1&0&0\\0&1&0\\1/3&1/3&1/3\\\end{bmatrix}$$ The both two first states are steady and the third is not.

Also as mentioned by @Stefan it is possible that $\lambda({\bf P}) =0$, making our matrix singular and impossible to invert. We will consider the matrices for which this does not happen.

My own work is limited to concluding that if any one vector has eigenvalue $1$, then there must exist a corresponding eigenvalue for the inverse which is the same, and this corresponding eigenvector for ${\bf P}^{-1}$ must correspond to the steady state eigenvector for $\bf P$.

But for the other eigenvectors of ${\bf P}^{-1}$, I don't know how to classify them. (or any other property which could be of interest)

Rodrigo de Azevedo
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mathreadler
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  • Do you want to place additional assumptions on $\mathbf P$? Not all stochastic matrices are invertible, e.g. take the matrix with first row equal to $1$ and $0$ oterhwise. Also the eigenvalue $1$ might not be "unique" in the sense that there are multiple linear independent eigenvectors, take for example the identity matrix, which is also stochastic. – Stefan Jul 16 '20 at 09:09
  • @Stefan Yep you are correct, I will update the question. – mathreadler Jul 16 '20 at 09:11

1 Answers1

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Suppose $P$ is an invertible, right-stochastic matrix. First, note the row sums of $P$ are equal to $1$, and as such so are the row sums of $P^{-1}$. (See this question.) Second, note that $P^{-1}$ is likewise right-stochastic if and only if $P$ is a permutation matrix (in which case $P = P^{-1}$). (See this other question). An implication of these two results is that each row in $P^{-1}$ contains at least one nonpositive element. That is to say that each row of $P^{-1}$ is necessarily an affine combination of the standard basis vectors, just not one that is a strict convex combination.

Let $\mathrm{conv}(M)$ denote the convex hull of the rows of matrix $M$. Geometrically, $\mathrm{conv}(P)$ is enclosed by the standard simplex (whose vertices are the standard basis vectors), and its inverse $P^{-1}$ is an enclosure of the standard simplex.

For example, if $$ P = \left[ \begin{array}{3} \color{blue}{0.60} & \color{blue}{0.30} & \color{blue}{0.10} \\ \color{orange}{0.15} & \color{orange}{0.70} & \color{orange}{0.15} \\ \color{green}{0.25} & \color{green}{0.00} & \color{green}{0.75} \end{array} \right], $$ then the geometry is as in the illustration below.

geometry

A notable special case is $P^{-1}$ with all off-diagonal elements nonpositive (and thus diagonal elements greater than or equal to $1$). In such a case, $P^{-1}$ is an $M$-matrix. Note that every invertible $M$-matrix with row sums equal to $1$ has a right-stochastic inverse.

brentertainer
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