Note the use of row vectors for point coordinates and that the matrix is written on the right.

The row vectors of P are the left eigenvectors of A.

Any such row vector is called a left eigenvector of .

Let this row vector be called .

An initial distribution is given as a row vector.

The row space of a matrix is the subspace spanned by its row vectors.

In particular, the row vectors of A are a basis for the null space of the corresponding matrix.

Notice that we are now using a column vector while the forward probabilities used row vectors.

It is relatively straightforward to see that the row vector corresponds, in the same way, to the reduction of modulo .

Then p is also a row vector and may present to another n by n matrix Q: