How to diagonalize specific 2x2 matrix

[fierce steeple]

I'm not completely unfamiliar with the starting steps. I subtracted lambda from the diagnals givin me the matrix
1 - 𝜆 0
10 -1 - 𝜆

then taking that to (1 - 𝜆)( -1 - 𝜆) - 0. foiling it out gives -1 + 1𝜆 - 1𝜆 + 𝜆^2 . i'm not sure how to proceed with the problem when 𝜆 is = to 0 ( unless i'm missing something which is really possible)

[quiet marten]

-1 + 1𝜆 - 1𝜆 + 𝜆^2 = 𝜆^2 -1 = 0

So, 𝜆 = 1, -1

Is this what you're looking for?

[dawn pilot]

Also, note that the eigenvalues of a triangular matrix are always on the diagonal of the matrix

(So 1 and -1 in this example)

[faint sparrow]

you can get some intuition for why this might be true by considering the 2x2 and 3x3 cases

formally you would need proof by induction but you don't need to, to be able to understand why

@fierce steeple

[unique horizon]

You don't need induction at all.
Just use the fact that the characteristic polynomial is just the product of the aii - X when the matrix is diagonal (as that's a known property of determinants)

[grizzled cradle]

once you have the eigenvalues, you find the associated eigenvectors

you want something tht looks like PDP^-1

D is the matrix with the eigenvalues on the diagonal