Linear Algebra Help: Proving Theorem 3 for nxn Matrices

[collinito]

Homework Statement

Explain why, if A is an n by n matrix,

and [1, . . . , 1] is a 1 by n matrix, then [1, . . . , 1]A will be a 1 by n matrix whose ith entry equals the sum of the entries in the ith column of A. Then use this idea to do the following problem.



1)Let P be an nxn probability matrix and let Z be an nx1 matrix. Prove that the sum of the entries in PZ is the same as that of the entries in Z.



2)Prove that the product of two probability matrices is a probability matrix.



3)Let P be an nxn probability matrix and let X=[1,1,...,1]^t be the nx1 matrix, all whose entries are 1. What is P^tX?(Try some examples if you are not sure.) How does it follow that det(P^t - I)=0 ?How does it follow that det(P-I)=0 how does this relate to proving theorem 3?



4)Prove that for any 2x2 probability matrix P the vector Y=[1,-1]^t is an eigenvector. Then prove theorem 3 in the 2x2 case.

Homework Equations

{Theorem 3; let P be a probability matrix with no entries equal to 0. Then there is a unique probability vector X such that PX=X. If D is any probability vector, then (lim->∞ PⁿD=X)}

The Attempt at a Solution

I don't know where to go with this all I know is an arbitrary 2-by-2 probability matrix will have the form [[a,b],[1-a,1-b]]

Note that Theorem 3 applies to most but not all probability matrices (so the assumptions about a and b have to be modified when proving Theorem 3). Note also that I need to prove not just that an equilibrium vector exists, but that it is unique. When proving the final claim in Theorem 3 (Pⁿ V₀ → X as n → ∞),

I believe it suffices to assume that the sum of the entries in V0 equals 1; the condition that those entries be non-negative shouldn’t get used in the proof.


This is all so confusing to me, I think I can handle this if I was given a direction to go. Could someone give me a hint as to how to handle this problem.

 

[lanedance]

ok, so what is your definition of a probability matrix, something about rows and/or columns with non-negative entries summing to one?

1) should follow from row/column properties of the probability matrix, starting in subscript notation (nx1 vectors are represented by only one index)

(PZ)_{i} = \sum_j P_{ij} Z_{j}