This is a more general case on Real Symmetric Matrices(RSM)

What is sufficient knowledge of a RSM's eigenvalue and eigenvector that is required to define a RSM uniquely?

For example, suppose we know that A is a 3 by 3 RSM, and it has eigenvalues t=2,t=-1,t=-1, and when t=2, its corresponding eigenvector is (1,1,1)^T

can we define A uniquely? And if so, how?

Thx for any reply or discussion