4. A probability transition matrix is a square matrix where...

4. A probability transition matrix is a square matrix where the sum of each row adds to one. One example would be A = egin{bmatrix} .5 & .4 & .1 \ .2 & .7 & .1 \ .9 & 0 & .1 end{bmatrix}. Show that ? = 1 is an eigenvalue of this matrix and find the corresponding eigenvector.

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Transcript

0:00 Hi.

00:01 In this problem, we're given a probability matrix a, which is shown here.

00:07 And a probability matrix is defined as a square matrix, such that each of the entries, the sum of the rows, equals 1.

00:16 So you can see that that is the case here.

00:19 So the first thing we want to do is show that the, that lambda equals 1 is an eigenvalue of a.

00:28 So remember, an eigenvalue lambda, sat satisfies the eigenvalue lambda satisfies that the determinant of a minus lambda i is equal to zero.

00:42 So if, uh, so if one is, uh, an eigen value, and this means that the determinant of a minus i should equal zero.

00:52 So here we have, well, i wrote out a minus lambda i.

00:56 And if you plug in, uh, lambda equals one here, um, you get point negative, or sorry, point negative five, negative, negative.

01:07 0 .5, negative 0 .5, you get negative 0 .3, you get negative 0 .9.

01:17 And then we calculate the determinant here.

01:20 So we have negative 0 .5 times the determinant of this minus 0 .4 times the determinant of this, and so on.

01:32 And this is just a calculation.

01:34 So i'll assume you should be able to do it.

01:36 But if you do this, you get this is the determinant.

01:39 And if you plug it into your calculator, you do in fact get zero.

01:44 I'm sorry, it's frozen a little bit...

Question

4. A probability transition matrix is a square matrix where the sum of each row adds to one. One example would be A = egin{bmatrix} .5 & .4 & .1 \ .2 & .7 & .1 \ .9 & 0 & .1 end{bmatrix}. Show that ? = 1 is an eigenvalue of this matrix and find the corresponding eigenvector.

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4. A probability transition matrix is a square matrix where the sum of each row adds to one. One example would be A = egin{bmatrix} .5 & .4 & .1 \ .2 & .7 & .1 \ .9 & 0 & .1 end{bmatrix}. Show that ? = 1 is an eigenvalue of this matrix and find the corresponding eigenvector.