Summary:
In several applications in mathematical sciences is required the computation
of the determinants and minors of matrices. The direct approach for evaluating
all the principal minors of a matrix by applying LU factorizations entails a
remarkable time complexity. Thus, analytical formulas will be useful to be
derived whenever it is possible. When we have matrices of special structure as
weighing matrices, this can be achieved.
In the present master thesis we concentrate our study on the evaluation of
minors for weighing matrices W(n, n-1) with zeros on the diagonal.
Specifically, we present known results for the evaluation of minors for
weighing matrices W(n, n-k), where n is even and kand we prove analogous propositions for weighing matrices W(n, n-1) with zeros
on the diagonal. Furthermore, for this specific category of matrices, we prove
the existence of an analytical formula for the evaluation of minors of order
n-r, where ntool of the first approach is the Determinant Simplification Theorem, while the
second one uses the orthogonality of the rows/columns.
Keywords:
Weighing matrices, Determinant–minors evaluation, Hadamard-equivalence, Schur complement, Determinant Simplification Theorem