Eigenvalues + shit (remedy a shitty textbook)

Help, my textbook sucks and has no examples on this type of problem.

Assume A is the weird eigenvalue symbol.
Find a 3x3 symmetric matrix that has eigenvalues A1=1,A2=3,A3=5 and the corresponding eigenvectors K1=(1;-1;1),K2=(1;0;-1),K3=(1;2;1).

I can easily do the reverse (find eigenvalues/vectors given a matrix), but have no clue where to even start here.  Any help?

Note that the 3 vectors are orthogonal -> try dumping them into a matrix (in this case horizontally)


U =
\left(
\begin{array}{ccc}
1 & -1 & 1 \\
1 & 0 & -1 \\
1 & 2 & 1 \end{array}
\right)


Because the vectors are orthogonal, U(K1) = [K1*K1,K2*K1,K3*K1] = [3,0,0], U(K2) = [0,2,0], U(K3) = [0,0,6]

Since U*([1,0,0]) = K1, U*([0,1,0]) = K2, etc
U*U(K1) = 3K1, U*U(K2) = 2K2, U*U(K3) = 6K3

Define D = [1/3, 3/2, 5/6]
Then A = U*DU

Why does this work?
K2 -U-> [0,2,0] = 2[1,0,0] -D-> 3[1,0,0] -U*-> K1, etc

Basically, all I'm doing is mapping your original basis vectors to another basis that is easier to work with, and mapping it back.

A is symmetric because A* = U*D*U = U*DU = A