Compound matrices: properties, numerical issues and analytical computations

Abstract

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD's) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD's of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.

A Mathematical Study of a Generalized SEIR Model of COVID-19

“…Let be a × complex matrix, be a × complex matrix and ≤ { , , } then ( ) = ( ) ( ). ⧫ Some other principal properties of compound matrices are given in [37][38][39][40] for ∈ (ℂ) and an integer, 1 ≤ ≤ : in particular, let ∈ (ℂ) and ≤ then we have :…”