To solve for the steady-state vector (or probability vector) of the matrix $P$, we need to follow these steps:

1. Steady-state equation: The steady-state vector $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ satisfies the equation: $P \mathbf{v} = \mathbf{v}$ where $P = \begin{bmatrix} 0.4 & 0.5 \\ 0.6 & 0.5 \end{bmatrix}$.

   This expands to: $\begin{bmatrix} 0.4 & 0.5 \\ 0.6 & 0.5 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$

2. Setting up the system: From this, we get two equations: $0.4 v_1 + 0.5 v_2 = v_1$ $0.6 v_1 + 0.5 v_2 = v_2$

3. Simplify the equations:

   • First equation: $0.4 v_1 + 0.5 v_2 = v_1$ becomes $-0.6 v_1 + 0.5 v_2 = 0$
   • Second equation: $0.6 v_1 + 0.5 v_2 = v_2$ becomes $0.6 v_1 - 0.5 v_2 = 0$

4. Solving the system: These two equations are: $-0.6 v_1 + 0.5 v_2 = 0$ $0.6 v_1 - 0.5 v_2 = 0$ These two equations are dependent, meaning they describe the same line in space. To find a unique solution, we add the condition that the sum of the components of $\mathbf{v}$ should be 1, since it is a probability vector: $v_1 + v_2 = 1$

5. Final solution: From the first equation, solve for $v_2$ in terms of $v_1$: $0.6 v_1 = 0.5 v_2 \quad \Rightarrow \quad v_2 = \frac{0.6}{0.5} v_1 = 1.2 v_1$ Now use the condition $v_1 + v_2 = 1$: $v_1 + 1.2 v_1 = 1 \quad \Rightarrow \quad 2.2 v_1 = 1 \quad \Rightarrow \quad v_1 = \frac{1}{2.2} = \frac{5}{11}$ Thus, $v_2 = 1.2 \times \frac{5}{11} = \frac{6}{11}$.

   Therefore, the steady-state vector is: $\mathbf{v} = \begin{bmatrix} \frac{5}{11} \\ \frac{6}{11} \end{bmatrix}$

Let me know if you'd like further details or have any questions.

Here are 5 related questions:

1. How does one check if a matrix is a valid stochastic matrix for steady-state calculations?
2. What is the significance of the steady-state vector in Markov chains?
3. Can a matrix have more than one steady-state vector?
4. How do eigenvalues relate to steady-state vectors?
5. What are the practical applications of steady-state vectors in probability theory?

Tip: The sum of the entries in a probability vector should always be 1, which helps in solving steady-state problems.