Hey jackGee.
Just to be clear, can you give a definition of symmetric mapping in mathematical form?
Let T : V -----> V be any linear mapping and a={v1,.....,vn} be any orthonormal basis of V
then the matrix A is symmetric iff <T(x),y> = < x ,T(y)> for x,y belong in V
I'm confused by the given basis b because usually I would find a set of eigenvectors and then use it to find an orthonormal basis
then show A is diagonalizable
Personally I'm wondering if you are dealing with an inner product with the following properties:
http://linear.axler.net/Chapter7.pdf