A generalization of a class of test matrices

Abstract:

We consider matrices of the following form: ${G_n}({a_1},{a_2}, \cdots ,{a_{n - 1}},{b_1},{b_2} \cdots {b_n}) =$ $({\beta _{i,j}}),1 \leqq i$, $j \leqq n$, where ${a_1}, \cdots ,{a_{n - 1}},{b_1}, \cdots ,{b_n}$ are constants and \[ {\beta _i}_{,j} = {b_j},{\text { }}j \geqq i;{\text { }}{\beta _{ij}} = {a_j},{\text { }}j < i.\] We deduce in analytic form the determinant, inverse matrix, characteristic equation, and eigenvectors of ${G_n}$. Knowing these properties enables us to generate valuable test matrices by appropriately selecting the order and elements of ${G_n}$.