Final answer:
Finding a steady state vector of a stochastic matrix requires setting up and solving a system of linear equations where the matrix multiplication leaves the vector unchanged, representing an equilibrium state of the system.
Step-by-step explanation:
When searching for a steady state vector of a stochastic matrix, one typically aims to find a vector that remains unchanged after application of the matrix. A steady state is when the initial transients die out and the system enters a condition of equilibrium. This involves using vector algebra and analytical methods to solve the equations set forth by the matrix. To find the steady state vector, one must set up a system of linear equations that reflects the condition that applying the matrix leaves the vector unchanged.
The process usually involves equating the original vector to the matrix transformation of that vector and solving for the vector components. Certain conservation laws or normalization conditions, such as those pertaining to momentum or wave functions, may assist in finding the solution. Once the equations have been set up and solved for the vector components, the resulting vector will represent the state in which the system can remain indefinitely or until an external change is enforced.