Answer:
To compute the stable matrix A, you can use the concept of stability in linear algebra. A matrix is stable if all its eigenvalues have absolute values less than 1. Here's how to determine if the given matrix A is stable:
First, find the eigenvalues of matrix A. The eigenvalues are the solutions to the equation det(A - λI) = 0, where λ represents the eigenvalues, and I is the identity matrix.
A = [[1, 0, 0.3], [0, 1, 0.2], [0, 0, 0.5]]
The equation det(A - λI) = 0 becomes:
|1-λ 0 0.3 |
|0 1-λ 0.2 |
|0 0 0.5-λ| = 0
Now, calculate the determinant:
(1-λ)((1-λ)(0.5-λ) - 0.2*0.3) - 0 - 0 = 0
Simplify the equation:
(1-λ)((0.5-λ) - 0.06) = 0
Now, solve for λ:
(1-λ)(0.5-λ-0.06) = 0
(1-λ)(0.44-λ) = 0
Expand:
0.44 - 0.44λ - λ + λ^2 = 0
λ^2 - 1.44λ + 0.44 = 0
Now, use the quadratic formula to find the eigenvalues:
λ = (-b ± √(b² - 4ac)) / (2a)
a = 1, b = -1.44, c = 0.44
λ₁ = (-(-1.44) + √((-1.44)² - 4*1*0.44)) / (2*1) = 0.6
λ₂ = (-(-1.44) - √((-1.44)² - 4*1*0.44)) / (2*1) = 0.84
Both eigenvalues, λ₁ and λ₂, have absolute values less than 1, which means the matrix A is stable.