Step by step
The question
Question: diagonalize A.
Diagonalizing means writing A as P times D times P inverse.
The columns of P are eigenvectors.
D is diagonal: the eigenvalues down the diagonal, 0s elsewhere.
1 Find the eigenvalues
As in the eigenvalue lesson: take λ off the diagonal of A.
Its determinant, down minus up, must be 0.
So (2 − λ)(2 − λ) − 0 × 1 = 0.
Multiply out the brackets.
Like terms together, the highest power first: λ² − 4λ + 4 = 0.
It factors, as in the factoring lesson.
So λ = 2.
2 Eigenvector for λ = 2
We take 2 off the diagonal: 2 − 2 = 0 and 2 − 2 = 0.
A row of 0s: the letter x is free.
That is why there are eigenvectors.
Row 1 as an equation: y = 0.
So y = 0.
We pick x = 1.
So an eigenvector for λ = 2 is (1, 0).
There is only one λ, and its eigenvectors are all numbers times (1, 0): one direction.
P needs two directions, so A cannot be diagonalized.
Let's look back.
First: find the eigenvalues.
Last: eigenvector for λ = 2.