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Data Analytics: Summaries on Strang's Linear Algebra - Markov Matrices PDF Print E-mail
Written by FemiByte   
Thursday, 28 February 2013 06:31

Markov Matrices

Properties of a Markov Matrix $A$:

  1. Every entry of $A$ is non-negative
  2. Every column of $A$ adds to 1

If $A$ is a positive Markov matrix (entries $a_{ij} > 0$, each column adds to 1), then $\lambda_1=1$ is larger than any other eigenvalue.
The eigenvector $x_1$ is the steady state: \[ u_k = x_1 + c_2(\lambda_2)^kx_2 + \cdots + c_n(\lambda_n)^k \\ x_n \rightarrow \text{ } u_\infty=x_1 \]

The following are the consequences of the properties of a Markov Matrix:

  1. $\lambda=1$ is an eigenvalue, with eigenvector $x_1>0$. This is the steady state.
  2. All other $|\lambda_i|<1$.

Perron-Frobenius Theorem

for $A>0$, all numbers in $Ax=\lambda_{max}x$ are strictly positive (+ve).

Consumption Matrix

Problem: Find a vector $p$ such that $p-Ap=y$ or $p=(I-A)^{-1}y$. Demand can be met only when $(I-A)^{-1}$ is non-negative. This is determined by the largest eigenvalue $\lambda_1$ of $A$ (which is +ve):

  • If $\lambda_1 > 1$ then $(I-A)^{-1}$ has negative entries.
  • If $\lambda_1 = 1$ then $(I-A)^{-1}$ fails to exist.
  • If $\lambda_1 < 1$ then $(I-A)^{-1}$ is non-negative as desired.

The formula for $(I-A)^{-1}$ is given by the geometric series for $(1+x)^{-1}, \text{ } |x|<1 $: \[(I-A)^{-1}=I+A+A^2+A^3+\cdots\] The series converges if all eigenvalues of $A$ have $|\lambda|<1$.

Last Updated on Thursday, 28 February 2013 15:33
 

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