Since U is a unitary matrix, S is a doubly stochastic matrix and we have \ tilde { a } = S \ tilde { \ lambda }.
22.
I . e . the elements of each column sum up to 1, so the matrix is a stochastic matrix ( for more details see the network analysis.
23.
:: : "'P "'is a stochastic matrix, which is an important fact to keep in mind for the rest of this discussion.
24.
This matrix is a stochastic matrix since sum of all its elements is equals to 1; i . e . \ sum _ j T _ { ij }.
25.
Additionally, every right stochastic matrix has an obvious column eigenvector associated to the eigenvalue 1 : The vector \ boldsymbol { 1 }, whose coordinates are all equal to 1.
26.
Another generalization that should be immediately apparent is to use a stochastic matrix for the transition matrices, and a probability vector for the state; this gives a probabilistic finite automaton.
27.
Intuitively, a stochastic matrix represents a Markov chain; the application of the stochastic matrix to a probability distribution redistributes the probability mass of the original distribution while preserving its total mass.
28.
Intuitively, a stochastic matrix represents a Markov chain; the application of the stochastic matrix to a probability distribution redistributes the probability mass of the original distribution while preserving its total mass.
29.
On the other hand, the Perron Frobenius theorem also ensures that every irreducible stochastic matrix has such a stationary vector, and that the largest absolute value of an eigenvalue is always 1.
30.
This can be seen by noting that \ mathcal { M } is by construction a stochastic matrix and hence has an eigenvalue equal to one as a consequence of the Perron Frobenius theorem.
