Find symmetric matrix containing no 0's, given eigenvalues I'm preparing for a final by going through the sample exam, and have been stuck on this: 
$$Produce\ symmetric\ matrix\ A ∈ R^{3×3},\ containing\ no\ zeros.\ \\ 
A\ has\  eigenvalues\ λ_1 = 1,\ λ_2 = 2,\ λ_3 = 3$$
I know $A = S^{-1}DS$, where A is similar to the diagonal matrix D, and S is orthogonal.
The diagonal entries of D are the eigenvalues of A. 
I also know that A & D will have the same determinant, eigenvalues, characteristic polynomial, trace, rank, nullity, etc. I am not sure where to go from here though. How cna A be found with only the two pieces of information? It seems like too little information is given...

Indeed too litle information is given and such a matrix is not unique.
I tried two run of the mill Orthogonal matrix, and it gave acceptable results.
The test didn't required you to find the unique matrix A satisfying these conditions.
Have a nice day :-)
