What is the second row of the matrix A, if A is...

What is the second row of the matrix A, if A is stochastic? $\begin{bmatrix} 0.2 & 0.7 \ ? & ? \end{bmatrix}$ Second row: 0.8 0.3 The steady state vector for A is $\begin{bmatrix} \\ \end{bmatrix}$ $\begin{bmatrix} 23 \ 31 \end{bmatrix}$ If $v = $, then $A^nv$ approaches $\begin{bmatrix} \\ \end{bmatrix}$ as $n$ gets large.

Transcript

00:01 Okay, so in this question we're asked to consider the matrix a that's incomplete, but we're just told it is a stochastic matrix.

00:12 And we're asked to first fill in the gaps, complete the matrix, second, find the steady state associated with the matrix, and then use our theory to find out what a n to the v converges to.

00:24 So the first observation, we need to recall what a stochastic matrix is, and this is simply that all entries are between 0 and 1, including 0 and 1, and the columns, whenever you sum a column, each column sums to 1, right? so for question a, to complete the matrix, 0 .4 we need to add up to 1, so 0 .6, 0 .3 needs to add up to 1, 0 .7, and we're done.

00:52 Now for part b, we have to remember what the steady state is.

00:56 So the steady state, what you proved in the lecture is that because this matrix, the columns add up to 1, 1 is an eigenvalue of the matrix.

01:05 So steady state is the one eigenvector associated with the eigenvalue 1, 1, and it satisfies the conditions.

01:17 Let's, if we say that this is a vector w, w1, w2, it satisfies w1, w2 are non -negative, and w1 plus w2 add up to 1, right? so this will be the eigenvector for the eigenvalue 1, whose entries add up to 1, and you know how to find this.

01:35 So this is a solution of a minus i, w1, w2 equals 0, 0.

01:42 So let's write the system.

01:45 So a minus the identity, this is 0 .4 minus 1, so minus 0 .6, 0 .3, 0 .6, we have in here, and then here we have 0 .7 minus 1, so minus 0 .3, w1, w2 equals 0, 0.

02:06 So of course, the two rows cancel each other out.

02:09 So what this gives us is minus 0 .6, w1 plus 0 .3, w2 equal to 0, which is the same as w2 equals 0 .6, w1 over 0 .3, which is 2, w1...

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