1 2 A 1
Find the steady state probability vector for the matrix. An eigenvector v of an n × n matrix A is a steady state probability vector when Av = v and the components of v sum to 1.
To find the steady state probability vector for matrix A = [[1/2, 1], [1/2, 0]], we need to follow the following steps:
Step 1: Compute the eigenvalues of matrix A. To do this, we need to solve the characteristic equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue.
det(A - λI) = 0 => det([[1/2-λ, 1], [1/2, -λ]]) = 0 => (1/2 - λ)(-λ) - 1/2 = 0 => λ^2 - (1/2)λ - (1/2) = 0
Solving this quadratic equation, we get λ1 = 1 and λ2 = -1/2.
Step 2: Compute the eigenvectors of matrix A corresponding to each eigenvalue. To do this, we need to solve the equation (A - λI)v = 0, where v is the eigenvector corresponding to the eigenvalue λ.
For λ1 = 1, we have:
(A - λ1I)v1 = 0 => ([[1/2, 1], [1/2, 0]] - [[1, 0], [0, 1]])v1 = 0 => [[-1/2, 1], [1/2, -1]]v1 = 0
Solving the system of linear equations, we get v1 = [2/3, 1/3].
For λ2 = -1/2, we have:
(A - λ2I)v2 = 0 => ([[1/2, 1], [1/2, 0]] - [[-1/2, 0], [0, -1/2]])v2 = 0 => [[1, 1], [1/2, 1/2]]v2 = 0
Solving the system of linear equations, we get v2 = [-1, 1].
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