8.2 Regular Stochastic Matrices

1. Regular Stochastic Matrix2. Stable Matrix and Distribution3. Properties of Regular Stochastic Matrix

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Regular Stochastic Matrix

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A stochastic matrix is said to be regular if some power has all positive entries.

Example Regular Stochastic Matrix

.6 .2 0 .5 0 1) ) )

.4 .8 1 .5 1 0a b c

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Which of the following stochastic matrices are regular?

a) All entries are positive so the matrix is regular.

Example Regular Stochastic Matrix (2)

20 .5 .5 .25)

1 .5 .5 .75b

2 30 1 1 0 0 1 0 1) and

1 0 0 1 1 0 1 0c

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All entries of the square are positive so the matrix is regular.

All powers will be one of the above two matrices so the original matrix is not regular.

Stable Matrix and Distribution If a stochastic matrix A has the properties

that 1. as n gets large, An approaches a fixed

matrix, and 2. any initial distribution approaches a

fixed distribution for large n, thenthe fixed matrix is called the stable

matrix of A and the fixed distribution is called the stable distribution of A.

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Example Stable Matrix and Distribution

In Jordan, 25% of the women currently work. The effect of maternal influence of mothers on their daughters is given by the matrix

Find the stable matrix and the stable distribution of A.

.6 .2.

.4 .8A

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Example Stable Matrix and Distribution (2)

1 13 3

2 23 3

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It appears that the powers are approaching

which is the stable matrix.

Example Stable Matrix and Distribution (3)

For the initial distribution,

However, for any initial distribution,

which is the stable

distribution.

1 1 1.253 3 3 .2 2 .75 2

3 3 3

1 1 13 3 3

2 2 1 23 3 3

pp

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Properties of Regular Stochastic Matrix

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Let A be a regular stochastic matrix.1. The powers An approach a certain matrix as n gets large. This limiting matrix is called the stable matrix of A.2. For any initial distribution [ ]0, An[ ]0 approaches a certain distribution. This limiting distribution is called the stable distribution of A.

Properties of Regular Stochastic Matrix (2)

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3. All columns of the stable matrix are the same; they equal the stable distribution.4. The stable distribution X = [ ] can be determined by solving the system of linear equations

sum of the entries of 1 .

XAX X

Example Properties Regular Matrices

Use the properties of a regular stochastic matrix to find the stable matrix and stable distribution of

.6 .2.

.4 .8A

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Example Properties Regular Matrices (2)

Solve

This gives

1.6 .2

..4 .8

x yx xy y

1.4 .2 0.4 .2 0.

x yx yx y

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Example Properties Regular Matrices (3)

The last equation is (-1) times the second equation so we can solve just the first two equations.

1 22 .4 1 .6

1 2

1 1 1 1 1 1.4 .2 0 0 .6 .4

11 01 1 1 320 1 20 13 3

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Example Properties Regular Matrices (4)

Therefore, the stable distribution and stable matrix, respectively, are

1 1 13 3 3 and .

2 2 23 3 3

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Summary Section 8.2 - Part 1

A stochastic matrix is called regular if some power of the matrix has only positive entries.

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Summary Section 8.2 - Part 2

If A is a regular stochastic matrix, as n gets large the powers of the matrix, An, approach a certain matrix called the stable matrix of A and the distribution matrices approach a certain column matrix called the stable distribution. Each column of the stable matrix holds the stable distribution. The stable distribution can be found by solving AX = X, where the sum of the entries in X is equal to 1.

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