$$

Find a good basis

Problem

Run the SageMath code below or simply click here.

load("https://raw.githubusercontent.com/jephianlin/LA-notebook/main/lingeo.py")

### code
set_random_seed(None)

n = 2
spec = random_int_list(n, 2)
D = diagonal_matrix(spec)
Q = random_good_matrix(n,n,n,2)
A = Q * D * Q.inverse()

pretty_print(LatexExpr("A ="), A)

pretty_print(
    LatexExpr(r"\beta = \Big\{"), 
    *[Q[:,i] for i in range(n)], 
    LatexExpr(r"\Big\}")
)

Replace None with your favorite number and run the code again. The outcome is a matrix

$A$. Find a basis

$\beta$ such that

$[f_A{]}_{\beta }^{\beta }$ is diagonal and check your answer with the output. When you are familiar with the process enough, try n = 3 .

This note can be found at Course website > Learning resources.