Knowing that there is a probability matrix M (where all columns add to 1) which when applied to a given vector P produces the same vector P, what is the best solution to find M? I can get my head around it for a 2x2 example but cannot work out a general solution for larger matrices.

My simple example is where we know our steady state vector P:


So with eigen value = 1 Looking for matrix M where M.P = P

Therefore (M - I)P = 0


This gives equations:

(a-1) 1/3 + b 2/3 = 0

a = 0.5 b

c 1/3 + (d-1) 2/3 = 0

c = 2 - 2d

and because it is a probability matrix we know:

c = 1-a and d = 1-b

We can substitute to find the values of M:


But my question is how do I find the solution for bigger vectors such as the following where M would be 4x4 or even bigger where M is 10x10 etc. What is the fastest way to compute the solution?



1 Answer 1


The first equation, $\frac13 (a-1) + \frac23 b = 0,$ is actually equivalent to $a = 1 - 2b.$ It does not necessarily imply that $a = \frac12b.$ It just happens that by setting $a=\frac15$ and $b = \frac25,$ both the actual condition implied by $(M-I)P$ and the additional condition $a = \frac12b$ are satisfied. But that is not the only solution.

For example, try $$ M = M_1 = \begin{pmatrix} \frac13 & \frac13 \\ \frac23 & \frac23 \end{pmatrix}. $$

This satisfies both $\frac13 (a-1) + \frac23 b = 0$ and $\frac13 c + \frac23 (d-1) = 0,$ but most importantly, its columns each sum to $1$ and it satisfies $M_1P = P.$ In fact, it satisfies $M_1v = P$ for any vector $v$ whose entries have the sum $1.$

A trivial example, of course, is $M=I$, since $IP = P.$

Moreover, if $M_1P = P$ and $M_2P=P$ then $(qM_1)P = qP$ and $(rM_2P)=P,$ so $(qM_1 + rM_2)P = (q+r)P.$ If also $q+r=1$ then $(qM_1 + rM_2)P = P,$ that is, the linear combination of matrices $qM_1 + rM_2$ satisfies the conditions you set for $M.$ In other words, any weighted average of two matrices that satisfy the conditions for $M$ (that is, each matrix is multiplied by a scalar, the sum of the scalars is $1,$ and after scalar multiplication the resulting matrices are added together) will also satisfy the requirements of $M.$ Since you can choose $q$ to be anything you want and then set $r = 1 - q$ to satisfy $q+r=1,$ this gives you infinitely many choices as long as you can find two different solutions for $M$ (which we have done).

For example, the matrix you found in the question is $\frac65 M_1 - \frac15 I.$ To derive this linear combination of matrices you simply set $q M_1 + r I$ equal to the matrix in the question and see if you can solve for $q$ and $r$. As it turns out, you can, and the result is $q=\frac65,$ $r=-\frac15.$ You can check the result by writing out the results of the scalar multiplications and matrix addition in $\frac65 M_1 - \frac15 I$ and comparing the final result with your original matrix.

As we can see from these examples, the choice of a matrix $M$ is not uniquely determined by the requirements that $M$ be a probability matrix and that $MP = P.$

A general approach to finding an $n\times n$ matrix $M$ for a vector $P$ of $n$ entries is to identify a non-zero entry in $P$ (there must be at least one of these). Suppose $p_k \neq 0.$ Then construct a set of $n(n-1)$ matrices of dimension $n\times n$ as follows:

For each $i$ and $j$ such that $1\leq i \leq n,$ $1\leq j \leq n,$ and $i\neq k,$ construct a matrix $B(j,i)$ with entries $b_{ji} = 1$ and $b_{jk} = \frac{1}{p_k}(m_{ji} p_i),$ and set all other entries of this matrix to zero.

Each such matrix $B(j,i)$ satisfies the condition $B(j,i)\,P = 0.$ These matrices are all independent of each other and span a vector space $\mathcal L$ of $n(n-1)$ dimensions. If $L$ is any matrix in that vector space, then $LP = 0.$

We can construct an affine space of matrices $\mathcal A$ by adding the identity matrix $I$ to each matrix $L$ in $\mathcal L.$ If $A$ is a matrix in $\mathcal A$ then for some $L \in \mathcal L,$ $$ AP = (I + L)P = P + LP = P + 0 = P,$$ and therefore $M=A$ is a solution of $MP=P.$

Another way to put it is, given that $p_k\neq 0,$ you can construct an $n\times n$ matrix $A$ by putting any real numbers you like everywhere except in the $k$th column and then setting $a_{j,k}$ so that $$ a_{j,k} = \frac{1}{p_k} (p_j - a_{j,1}p_1 - \cdots - a_{j,k-1}p_{k-1} - a_{j,k+1}p_{k+1} - \cdots - a_{j,n}p_n). $$ The independent choices you make in each of the $n(n-1)$ entries other than the $k$th column give you a space of $n(n-1)$ dimensions. As long as $P\neq 0,$ however, $M=0$ is not a solution to $MP = P$ and therefore the space is an affine space rather than a vector space.

But this only shows that a matrix in this affine space is a solution to the equation $MP=P.$ Not all matrices in the space are probability matrices. The requirement that $M$ must be a probability matrix imposes more constraints.

To help discuss the implications of these constraints, suppose that a particular probability matrix $A$ can be written $$ A = I + r_{11} B(1,1) + \cdots + r_{1n} B(1,n) + \cdots + r_{n1} B(n,1) + \cdots + r_{nn} B(n,n), $$ that is, as the sum of $I$ and a linear combination of the matrices $B(j,i)$ previously described where $j$ runs from $1$ to $n$ and $i$ runs from $1$ to $n$ excluding $k.$

One set of constraints says that every entry of $A$ must be in $[0,1].$ For each $i \neq k,$ we have $a_{ii} = 1 + r_{ii},$ so the constraint implies that $-1 \leq r_{ii} \leq 0.$ But for $i\neq k$ and $i\neq j$ we have $a_{ji} = r_{ji},$ so the constraint implies that $0 \leq r_{ji} \leq 1.$

A further constraint is that the sum of each column must be $1.$ But the matrix $I$ has entry $1$ in each column, so the sum of entries in column $i$ of $A$ (where $i\neq k$) is $$ 1 = 1 + r_{1i} + \cdots + r_{ni}, $$ so the sum of all the coefficients $r_{ji}$ must be zero. This implies that the contributions of the matrices $B(j,i)$ to the sum of entries in column $k$ is also zero, so we find that the $k$th column also sums to $1$ without imposing any further constraint. However, the constraints on the sum of $r_{ji}$ for every $i$ such that $i\neq k$ reduces the dimension of the solution by $n - 1.$ The remaining solution is a convex subset (bounded by the conditions $-1 \leq r_{ii} \leq 0$ and $0 \leq r_{ji} \leq 1$ for $i\neq j$) within an $(n-1)^2$-dimensional affine space.

When $n=2$ the solution is a convex subset of a space of dimension $(2-1)^2 = 1.$ So it is no coincidence after all that the matrix found in the question is a linear combination of $I$ and the matrix $M_1$ found earlier in this answer. All solutions to that particular problem have that property.

For the vector $P = \begin{pmatrix} 1/3 \\ 2/3 \end{pmatrix},$ the solution space of $M$ consists of all matrices that can be written $$ \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + r \begin{pmatrix} -2 & 1 \\ 0&0 \end{pmatrix} - r \begin{pmatrix} 0&0 \\ -2 & 1 \end{pmatrix} $$ where $0 \leq r \leq 1.$

In particular, $$ \begin{pmatrix} 0.2 & 0.4 \\ 0.8 & 0.6 \end{pmatrix} = \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + \frac25 \begin{pmatrix} -2 & 1 \\ 0&0 \end{pmatrix} - \frac25 \begin{pmatrix} 0&0 \\ -2 & 1 \end{pmatrix} $$ and $$ \begin{pmatrix} \frac13 & \frac13 \\ \frac23 & \frac23 \end{pmatrix} = \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + \frac13 \begin{pmatrix} -2 & 1 \\ 0&0 \end{pmatrix} - \frac13 \begin{pmatrix} 0&0 \\ -2 & 1 \end{pmatrix}. $$

  • $\begingroup$ Thanks for this David. So does that mean that, assuming we are told there is a 9x9 matrix M that when multiplied by a known vector P will give the same vector P, it is not possible to calculate the matrix M? Is this because, as per your example, there are multiple matrices M that would do the job? Is there a way to find any/all of the possible solutions for M? $\endgroup$
    Apr 10, 2019 at 9:47
  • $\begingroup$ Yes, there is an infinite number of possible matrices $M$ for each vector $P$ with $n$ entries, $n>1.$ The set of these matrices can be completely described as a vector space, so I have added this to the answer. $\endgroup$
    – David K
    Apr 10, 2019 at 11:42
  • $\begingroup$ I'm confused, how can the set of matrices such that $P=MP$ for a fixed $P$ be a vector space? It cannot possibly be closed under either scalar multiplication or addition. What you want is the analogue of a vector space where linear combinations are replaced by convex combinations. $\endgroup$
    – Ian
    Apr 10, 2019 at 11:52
  • $\begingroup$ @David I'm not sure I follow the explanation of how to calculate a possible M. Could you give an example? I also don't follow the link between the infinite solutions you mentioned - I see that you linked my solution to your matrix M1 - but don't know how you formed this relationship. $\endgroup$
    Apr 10, 2019 at 14:30
  • $\begingroup$ I just happened to notice the relationship; that's how I found out about it. It helps if you can do small matrix calculations like this in your head. But I've given procedures for how you can either deduce or confirm the fact with pencil and paper. It isn't guaranteed that three arbitrary solutions to $MP=P$ would have this relationship; just a lucky coincidence in this case. $\endgroup$
    – David K
    Apr 10, 2019 at 21:25

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