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What are the relations between a matrix H and its transpose H^T? I am not asking about the relations between the coefficients, I am asking the relations as linear maps (H: F^m->F^n; H^T: F^n->F^m). I am not sure exactly how I should pose the question actually, but I am thinking there is some deeper relations than between their coefficients, like for example, can we say something about the kernel and image of H^T if we know something about the kernel and image of H? What can we say about HH^T: F^n->F^n and H^TH: F^m->F^m?
(F is an arbitrary field)
(F is an arbitrary field)