Are transition matrices Square?

A transition matrix consists of a square matrix that gives the probabilities of different states going from one to another.

What is a valid transition matrix?

Regular Markov Chain: A transition matrix is regular when there is power of T that contains all positive no zeros entries. You can also look at it as irreducible matrix with at least one element in the main diagonal not equal to zero.

What does it mean if a matrix is not square?

For a non-square matrix with rows and columns, it will always be the case that either the rows or columns (whichever is larger in number) are linearly dependent. Hence when we say that a non-square matrix is full rank, we mean that the row and column rank are as high as possible, given the shape of the matrix.

Are Markov matrices Square?

These transition matrices will also always be square (i.e., same number of rows as columns) since we want to keep track of the probability of going from every state to every other state, and they will always have the same number of rows (and same number of columns) as the number of states in the chain.

How do you determine whether the Markov chain with that transition matrix is regular?

To determine if a Markov chain is regular, we examine its transition matrix T and powers, Tn, of the transition matrix. If we find any power n for which Tn has only positive entries (no zero entries), then we know the Markov chain is regular and is guaranteed to reach a state of equilibrium in the long run.

Does inverse exist for non-square matrix?

Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse.

Do all Markov matrices have eigenvalue 1?

+ pn = 1. A Markov matrix A always has an eigenvalue 1. All other eigenvalues are in absolute value smaller or equal to 1. Because A and AT have the same determinant also A − λIn and AT − λIn have the same determinant so that the eigenvalues of A and AT are the same.