About Markov matrix


In W4โ€™s lab, the lecturer says โ€œPredicting probabilities in ๐‘‹๐‘š when ๐‘š is large you can actually look for an eigenvector corresponding to the eigenvalue 1 , because then you will get ๐‘ƒ๐‘‹=๐‘‹โ€. I donโ€™t quite understand it, does it mean PXinf should equal to Xinf ? Why this is the case?

Hi @JerryLee!

@esanina, can you give a hand here? Thanks!

@JerryLee thank you for the question. Have a look at the equation PX_{m-1} = X_{m}. Matrix P has an eigenvalue 1 for sure, so if X is an eigenvector corresponding to the eigenvalue 1, then PX = X. Now have a look at both of those equations:
PX_{m-1} = X_{m}
PX = X
You are interested in the probabilities of the browser after infinite steps of navigation, so for large m, such a vector X_{m-1} which will not change much with the transformation P. And that corresponds to the equation PX = X which is the same as an equation for an eigenvector corresponding to the eigenvalue 1.

Hi Esanina,

Why Xm wonโ€™t change much with P when m is large?


Hello @JerryLee,

I donโ€™t mentor this course so I have no access to the lab, but with the info in this thread, I can share a few points as the direction for you to figure the rest out yourself.

Considering a positive Markov matrix P,

  1. Any vector x may be written as a linear combination of the eigen vectors of P

  2. Eigen values of P are -1 < lambda <= 1.

  3. Applying P to x shrinks all of xโ€™s eigen vector components except the component with eigen value of 1. (Why shrink? Think about the eigen values.)

  4. Repeatedly applying P keeps shrinking.

  5. Infinitely repeatedly applying P zeros out all components except for the component with the eigen value of 1.

Please try to follow these ideas and derive the equation in question.

Good luck, and cheers,