Donīt know if the descriptive title is misleading but here goes.

Show that for every matrix with a range there exists a matrix which is symmetric and has a range .

This is what I have:

The matrix is a square matrix and also symmetric because .

The part with range is still confusing. Now this should be true

but how should I pursue ? Im thinking about Orthogonal matrices. Because if is symmetric which it is, then there exists an orthogonal matrix such that where is a diagonal matrix with the eigenvalues of in the diagonal.

any help appreciated