- #1
evilpostingmong
- 339
- 0
Homework Statement
Prove or give a counterexample: the product of any two selfadjoint
operators on a finite-dimensional inner-product space is
self-adjoint.
Homework Equations
The Attempt at a Solution
I'd say that if we let a diagonal matrix represent T (after all, its transpose representing
T*=the matrix representing T) and multiply it by a diagonal matrix representing
the transformation S, then we'd end up with a diagonal matrix as a product.
So the product is self adjoint since all diagonal matrices are equal to their
transposes.
Another case is with an nxn matrix where all entries are equal. This matrix represets T and its
transpose is T*. Its matrix = its transpose so itself adjoint. Now multiplying it with another nxn matrix
representing S with all entries equal to each other would obviously produce a matrix with all entries
equal to each other. Or multiplying the matrix for T with a diagonal matrix would produce a diagonal matrix.
Last edited: