Note: the last column of

zeros in A-I_3 doesn't

belong.

NOTE: in fall 2018, we did not cover orthogonal matrices,

so you would be expected to just find

P=[v_1 v_2 v_3] and D (the diagonal matrix with diagonal

entries 1, -2,-2 and note that A=PDP^{-1} .

To find an orthogonal matrix, one needs to find an

orthonormal basis of the eigenspace for lambda _1=1 and an

orthonormal basis of the eigenspace for lambda_2=-2 and

that's what they are doing at the bottom of the page and on

the next page.