# Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors

## Problem 34

**(a)** Let

\[A=\begin{bmatrix}

a_{11} & a_{12}\\

a_{21}& a_{22}

\end{bmatrix}\]
be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$.

(Such a matrix is called (right) * stochastic matrix* (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix).)

Then prove that the matrix $A$ has an eigenvalue $1$.

**(b)** Find all the eigenvalues of the matrix

\[B=\begin{bmatrix}

0.3 & 0.7\\

0.6& 0.4

\end{bmatrix}.\]

**(c)** For each eigenvalue of $B$, find the corresponding eigenvectors.

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## Hint.

- For (a), consider the vector $\mathbf{x}=\begin{bmatrix}

1 \\

1

\end{bmatrix}$. - For (b), use (a) and consider the trace of $B$ and its relation to eigenvalues.

For this relation, see the problem Determinant/trace and eigenvalues of a matrix.

## Solution.

### (a) Prove that the matrix $A$ has an eigenvalue $1$.

Let $\mathbf{x}=\begin{bmatrix}

1 \\

1

\end{bmatrix}$ and we compute

\begin{align*}

A \mathbf{x}=\begin{bmatrix}

a_{11} & a_{12}\\

a_{21}& a_{22}

\end{bmatrix}

\begin{bmatrix}

1 \\

1

\end{bmatrix}

=\begin{bmatrix}

a_{11}+a_{12} \\

a_{21}+a_{22}

\end{bmatrix}

=\begin{bmatrix}

1 \\

1

\end{bmatrix}

=1\cdot \mathbf{x}.

\end{align*}

This shows that $A$ has the eigenvalue $1$.

### (b) Find all the eigenvalues of the matrix $B$.

Note that the matrix $B$ is of the type of the matrix in (a).

Thus the matrix $B$ has the eigenvalue $1$. Since $B$ is $2$ by $2$ matrix, it has two eigenvalues counting multiplicities. To find the other eigenvalue, we note that the trace is the sum of the eigenvalues.

Thus we have

\[\tr(B)=0.3+0.4=1+\lambda,\]
where $\lambda$ is the second eigenvalue. Hence another eigenvalue is $\lambda=-0.3$.

### (c) For each eigenvalue of $B$, find the corresponding eigenvectors.

By solving $(B-I)\mathbf{x}=\mathbf{0}$ and $(B+0.3I)\mathbf{x}=\mathbf{0}$, we find that

\[\begin{bmatrix}

1 \\

1

\end{bmatrix}t \quad \text{ and } \quad \begin{bmatrix}

-7 \\

6

\end{bmatrix}t\]
for any nonzero scalar $t$ are eigenvectors corresponding to eigenvalues $1$ and $-0.3$, respectively.

## Comment.

For some specific matrices, we can find eigenvalues without solving the characteristic polynomials like we did in part (b).

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## 1 Response

[…] The matrix $A$ is called a stochastic matrix (or Markov matrix, probability matrix). For the definition of these terminology and a similar problem, see problem Stochastic matrix (Markov matrix) and its eigenvalues and eigenvectors. […]