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 $A = egin{bmatrix}.5 & .4 & .1\.2 & .7 & .1\.9 & 0 & .1end{bmatrix}$. Show that $lambda = 1$ is an eigenvalue of this matrix and find the corresponding eigenvector.

The transition matrix in Example 5 has the property that both its rows and its columns add up to $1 .$ In general, a matrix $A$ is said to be doubly stochastic if both $A$ and $A^{T}$ are stochastic. Let $A$ be an $n \times n$ doubly stochastic matrix whose eigenvalues satisfy $$ \lambda_{1}=1 \quad \text { and } \quad\left|\lambda_{j}\right|<1 \text { for } j=2,3, \ldots, n $$ Show that if $\mathbf{e}$ is the vector in $\mathbb{R}^{n}$ whose entries are all equal to $1,$ then the Markov chain will converge to the steady-state vector $\mathbf{x}=\frac{1}{n} \mathbf{e}$ for any starting vector $\mathbf{x}_{0} .$ Thus, for a doubly stochastic transition matrix, the steady-state vector will assign equal probabilities to all possible outcomes.

Eigenvalues

Diagonalization

Question

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