Math and science::Algebra

Markov matrices and their eigenvectors. An NZ-AU immigration example.

What is the population of Australia and New Zealand after 100 years of the following migration pattern? Starting with 1 million people in Australia and no one in New Zealand.

To start things off, the transition is described by:

[\begin{matrix} {Australia}_{t + 1} \\ {New Zealand}_{t + 1} \end{matrix}] = [\begin{matrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{matrix}] [\begin{matrix} {Australia}_{t} \\ {New Zealand}_{t} \end{matrix}]

Continue on the reverse. How does $X V X^{- 1}$ come up?

The answer is the result of the following expression:

{[\begin{matrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{matrix}]}^{100} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}]

To calculate the result, use eigenvalues and eigenvectors. The transition matrix has eigenvalues 1.0 and 0.5 corresponding to eigenvectors:

[\begin{matrix} 0.6 \\ 0.4 \end{matrix}] and [\begin{matrix} - 1 \\ 1 \end{matrix}]

We can express the starting population as a linear combination of the eigenvectors:

[\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}] = [\begin{matrix} 600, 000 \\ 400, 000 \end{matrix}] + [\begin{matrix} 400, 000 \\ - 400, 000 \end{matrix}]

Then,

{[\begin{matrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{matrix}]}^{100} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}]

becomes:

{1.0}^{100} [\begin{matrix} 600, 000 \\ 400, 000 \end{matrix}] + {0.5}^{100} [\begin{matrix} 400, 000 \\ - 400, 000 \end{matrix}]

Which is

[\begin{matrix} 600, 000 \\ 400, 000 \end{matrix}]

General approach

The above intuitive approach is an application of the somewhat opaque symbolic expression $X V X^{- 1}$ . The steps are:

Express the starting population as a linear combination of eigenvectors.
Apply the transition matrix to the eigenvectors.
Express the result as a linear combination of eigenvectors.

Repeat the solution from this perspective, letting $A$ be the transition matrix, and $X$ be the matrix of eigenvectors and $V$ be the diagonal matrix of eigenvalues. We will set $X$ to be:

[\begin{matrix} 600, 000 & 400, 000 \\ 400, 000 & - 400, 000 \end{matrix}]

We are trying to calculate:

result = A^{100} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}]

By eigenvector decomposition, we have:

result = X V^{100} X^{- 1} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}]

First, calculate $X^{- 1} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}]$ :

X^{- 1} [\begin{matrix} 1, 000, 000 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \end{matrix}]

Then multiply by $V^{100}$ :

V^{100} [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} {1.0}^{100} \\ {0.5}^{100} \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}]

Finally, get back to the original basis:

X [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 600, 000 & 400, 000 \\ 400, 000 & - 400, 000 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 600, 000 \\ 400, 000 \end{matrix}]

Source

https://www.flickr.com/photos/archivesnz/18288648358