I helped a buddy of mine do some MATLAB homework where you had to orthogonalize a matrix via Gram-Schmidt. I wrote a test function called
isorthog that returned true or false based on whether the (orthongonalized) matrix times its transpose was equal to the identity matrix of the same size. His professor told him that this check is unnecessary. I think it is, because sometimes it would test false.
Can all $n \times n$ matrices be orthogonalized?
If so, can someone provide a hint as to how to prove it? I DO NOT want a full solution just 1 or 2 pointers to help me get started. Thanks!