1. The problem statement, all variables and given/known data This is for linear algebra/matrix: Orthogonally diagonalize this matrix A by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D A = [ 1 2 2 ] [ 2 1 2 ] [ 2 2 1 ] 2. Relevant equations (A - [tex]\lambda[/tex]I ) = 0 3. The attempt at a solution D = [5 0 0 ] [0 -1 0 ] [0 0 -1 ] characteristic equation : -[tex]\lambda[/tex]3 + [tex]\lambda[/tex]2 + 9[tex]\lambda[/tex] + 5 = 0 [tex]\lambda[/tex] = 5, -1, -1 (I got these after factoring the characteristic equation) when [tex]\lambda[/tex] = 5, I got v1 = [ 1 1 1 ] Then I'm almost done but I got stuck when trying to find v2 and v3 when [tex]\lambda[/tex] = -1 because when I tried to do it, it turned out weird (it turned into a zero matrix!): [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] So I think it means that x1 , x2 and x3 are all free variables for v2 and v3 , but if that's the case, then how can I make v1 v2 v3 into an orthogonal matrix if they're not independent?!?! I almost got it but I've no idea what to do now! Does this mean that it is not possible to orthogonally diagonalize it?