In both Puzzles 4 and 5, we did not have to use the self-adjointness of , so it could have been an infinitesimal stochastic matrix, following the same definition of irreducibility. So, we get the required result.

]]>If is a conserved quantity for , then , implying that for . Now, as is a diagonal matrix, therefore for all .

If is an irreducible Dirichlet operator, then from Puzzle 4, there exists a natural number m such that for every with .

Now, assume that is not a constant. Then there exists at least one pair , for which . Then from the previous paragraph, for all , which contradicts the fact that is an irreducible Dirichlet operator.

Conversely, if we assume that the only s for which , are constant, then there has to be at least one for which ; otherwise need not be equal to .

]]>For checking connectedness, we are concerned only with the non diagonal terms.

Since , and have the same non-diagonal terms, if for each with , there exists a natural number m such that , then and vice versa.

So, is irreducible iff is irreducible.

]]>For Puzzle 3:

Given that T is a nonnegative matrix, we can associate a graph with it, as was done in the post.

Notice that .

If is non zero, it means that neither nor is zero, i.e. there exists an edge from vertex to and an edge from to . So , if non zero, corresponds to a directed 2-edge path from to .

Since, all elements of are nonnegative, is zero only if there is no directed 2-edge path from to . If it is non-zero, there is at least one directed 2-edge path connecting and .

Now . If non zero, there is at least one directed 2-edge path from to and a directed edge from to ; therefore a directed 3-edge path from to .

Through induction, is non zero, if there is a directed m-edge path from to .

Since, T is irreducible iff there is a directed path from every vertex to every other vertex in , we get the result that T is irreducible iff there exists a natural number m such that , so that there is at least one directed path connecting any pair of vertices.

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