Question

Can someone help me check if this matrix is diagonalisable?

`[(4,-5), (2,3)]`

Answer 1

See below

Explanation:

`[(4,-5), (2,3)]`

From the characteristic equation:

`lambda^2 - Tr(M) lambda + det(M) = 0`

`implies lambda^2 - 7 lambda + 22 = 0`

Solve the quadratic for eigenvalues:

• `lambda_(1,2) = (7 pm i sqrt(39))/2`

These will deliver 2 distinct eigenvectors so you can diagonalise .

For eigenvectors `bb alpha_(1,2)` , use `M bb alpha_i = lambda_i bb alpha_i`:

`[(4,-5), (2,3)] bb alpha_1 = (7 + i sqrt(39))/2 bb alpha_1`

• `implies 4 x - 5 y = (7 + i sqrt(39))/2 x`

• `5y = (8 - 7 - i sqrt(39))/2 x`

`implies y = (1 - i sqrt(39) )/10 x`

A first eigenvector is:

`bb alpha_1 = [(1),((1 - i sqrt(39) )/10)]`

For the second:

`[(4,-5), (2,3)] bb alpha_2 = (7 - i sqrt(39))/2 bb alpha_2`

• `4 x - 5 y = (7 - i sqrt(39))/2 x`

`implies y = (1 + i sqrt(39) )/10 x`

A second eigenvector is:

`bb alpha_2 = [(1),( (1 + i sqrt(39) )/10)]`