eigenvectors are known, how to find the symmetric matrix?

There are probably multiple ways to do this. The one that comes to mind is the expansion of A in terms of eigenvectors and eigenvalues:

A = lambda1 * v1 v1' + lambda2 * v2 v2' + lambda3 * v3 v3

The prime notation means transpose. A matrix uv' is the "outer product" of u and v: The ij-th element is the i-th element of u times the j-th element of v.

Thinking about it a bit, that expansion only works if the eigenvectors are orthonormal. Yours are orthogonal but not normalized. You need to normalize v1 and v3 by dividing by sqrt(2).

Edit: Another thought. A can be written in the form UBU' where B is a diagonal matrix whose elements are the eigenvalues and U is an orthogonal matrix whose columns are the eigenvectors. This is the theory of "diagonalizing a matrix". But actually that's equivalent to what I wrote above.