Question

Find a basis for the space V of real nxn symmetric matrices. Prove your claim.
Then, find a basis for the for the complimentary space, that is, the vector space W such that every nxn real matrix is a sum of a vector v E V and w E W, and that V? W = {ô} ({o vector})?

Answer 1

Let {e_i} (i=1,.....,n) be an orthonormal basis of n-dimensional unit vectors. Then you can make a basis for symmetric matrices by

E_{ij} = 1/2 (e_i e_j^T + e_j e_i^T)

Clearly these matrices are all symmetric and orthogonal(the inner product ios the trace of the product.)

To prove this is a basis, use the fact that the rows and columns of any matrix A are
vecors and the properties of the e_i as a basis for vectors.