Linear Algebra; Stochastic matrix and Steady State vectors

In summary, the conversation is discussing how to show that every 2x2 stochastic matrix has at least one steady-state vector. The matrix is defined as P = |1-a b| |a 1-b|, where a and b are constants between 0 and 1. The participants mention that there are two linearly independent steady-state vectors if a = b = 0, otherwise there is only one. The solution involves finding a nonzero solution to the equation (P-I)x=0, where I is the 2x2 identity matrix.
  • #1
MustangGt94
9
0

Homework Statement



Question: 18. Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written as

P = |1-a b |
| a 1-b |

where a and b are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = b = 0. Otherwise, there is only one.)

The Attempt at a Solution



I am guessing that they want me to show that the above matrix has a solution to the equation Px = x so that the steady state vector exists since there is a solution. My solution was that since a and b are other constants between 0 and 1 the matrix would have a solution?

Thank you for the help!
 
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  • #2
Hi Mustang,

[tex]P = \begin{pmatrix} 1-a & b\\ a & 1-b\end{pmatrix}[/tex]

Consider that if Px = x, then Px - x = 0, i.e., (P - I)x = 0, where I is the 2x2 identity matrix. As a steady state vector (just a certain type of "eigenvector", if you are familiar with the term) is necessarily nonzero, recall how one can use the determinant to determine where the system Ax=0 has a nonzero solution.
 
Last edited:
  • #3
Ah! I see what you mean, thanks a lot Unco, much appreciated!
 
1.

What is Linear Algebra and why is it important in science?

Linear Algebra is a branch of mathematics that deals with linear equations, linear transformations, and vector spaces. It is important in science because it provides a useful framework for representing and solving systems of equations, which are commonly used in modeling real-world phenomena.

2.

What is a Stochastic Matrix and how is it used?

A Stochastic Matrix is a square matrix in which all entries are non-negative and each row sums to 1. It is commonly used in probability theory to represent the transition probabilities between states in a Markov Chain. It can also be used to model the behavior of random systems in various fields such as economics, biology, and physics.

3.

What is a Steady State vector and how is it calculated?

A Steady State vector, also known as a stationary vector, is a vector that remains unchanged under a given transformation. In the context of Stochastic matrices, it represents the long-term behavior of a system. It can be calculated by finding the eigenvector corresponding to the eigenvalue 1 of the Stochastic matrix.

4.

How is Linear Algebra used in machine learning and data analysis?

Linear Algebra plays a crucial role in machine learning and data analysis as it provides a powerful framework for representing data and performing operations on large datasets. It is used in techniques such as Principal Component Analysis, Singular Value Decomposition, and Linear Regression, which are commonly used in data analysis and machine learning algorithms.

5.

What are some real-world applications of Linear Algebra and Stochastic matrices?

There are countless real-world applications of Linear Algebra and Stochastic matrices, some of which include image and signal processing, financial modeling, population dynamics, and game theory. They are also used in various engineering fields such as control systems, circuit analysis, and robotics.

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