Inverse matrix and properties of double determinant over quaternion field (Q1179420)

Let \(A,B\in Q_{n\times m}\), this being the set of \(n\times m\) matrices over the quaternions. With stars * indicating transpose conjugation, define \(\| A\|=\det(A^*A)\). Contents of paper: \(\bullet\) If \(n=m\), then \(\| A^*\|=\| A\|\) and \(\| AB\|=\| A\| \| B\|\). \(\bullet\) For general \(A\), \(\| A\|\geq 0\). \(\bullet\) If \(n=m\), then \(A\) is invertible if \(\| A\|\neq 0\); \(A^{-1}\) is given explicitly. \(\bullet\) Hadamard's determinant inequality is derived from the more general \[ \|(\alpha_ 1,\ldots,\alpha_ s,\alpha_{s+1},\ldots,\alpha_ m)\|\leq\|(\alpha_ 1,\ldots, \alpha_ s)\|\|(\alpha_{s+1},\ldots,\alpha_ m)\|, \] the \(\alpha\)'s being column vectors in \(Q^ n\). \(\bullet\) Right and left linear independence as well as rank concepts are studied. Some of the proofs, as far as checked (essentially) correct, are quite technical; perhaps necessarily so since fundamental things like \(\hbox{det}{a \lambda a\choose b \lambda b}=0\) are usually false for quaternions. Acta Math. Sinica 16, No. 1, 1-10 (1990) contains the author's earlier investigations.