Hadamard定理在四元数体上的推广

AN EXTENSION OF HADAMARD THEOREM OVER THE SKEW-FIELD OF QUATERNIONS

Let K be any skew-field with central field F. A matrix A=(a_ij)n×n over K is called centralized if the characteristic matrix λI-A can be reduced by some elementary transformations into the following diagonal form:such that are all manic polynomials over F. The determinant of a centralized matrix A=(a_ij)n×n may be defined by (1) asfollows: and then the famous theorem of Hadamard can be generalized as in the following: Theorem. If A=(a_ij)n×n is a non-singular centralized matrix over the skewfield of quaternions, thenand the equality sign holds if and only if the columns of A are all muturally orthogonal. This theorem may be proved by the following three lemmas: Lemma 1. If A is a centralized n-rowed square matrix of quaternions, then sois and Lemma 2. For any n-rowed square, matrix A of quaternions, the. following two matrices are always centralized:and Lemma 3. If A is a centralized matrix of quaternions, then we have