Prove That a Real Matrix and Its Gram Product Have Equal Rank

For a real m-by-n matrix A, prove rank(A)=rank(A*A^T). ### Constraints and Clarifications A^T denotes transpose and the entries are real. No full-rank or square-matrix assumption is allowed. Explain what changes over complex numbers, where conjugate transpose may be required. ### What a Strong Answer Covers Relate the null spaces of A^T and A*A^T, use a nonnegative squared norm, and apply rank-nullity in the correct ambient dimension. ### Follow-up Questions - Why does rank(A*A^T)<=rank(A) alone not complete the proof? - Can the statement fail over complex entries if plain transpose is used?