On the determinant of general pentadiagonal matrices

Abstract

In this paper we consider square matrices with two sub-diagonals and two super-diagonals. We provide an algorithm to transform such matrices (by multiplying them with suitable matrices) to tridiagonal matrices. It is known that tridiagonal matrices can be transformed to diagonal ones (again by multiplying them from both sides by suitable matrices). Thus pentadiagonal matrices can be transformed to diagonal ones and in this way their determinants can be calculated. Two examples show how our method works.

Determinants of some pentadiagonal matrices

“…are the same as in [8,9] however the transformation rules for the entries are different. In those papers during the first (and subsequent) processes the diagonal vectors l, r just shortened while the diagonal vectors L, R transformed.…”

“…In [8] we developed a method to reduce the determinant of k, ℓ-pentadiagonal matrices to tridiagonal determinants provided that k + ℓ ≥ n + 1. If k ≥ (n + 1)/3 then by this method in [9] we determined the determinants of general, Toeplitz and imperfect Toeplitz k, 2k-pentadiagonal matrices.…”

“…If the condition k + ℓ ≥ n + 1 is not satisfied then the method given in [8] is not applicable in general. However, if ℓ = 2k then with some modification it is applicable.…”

Products and inverses of multidiagonal matrices with equally spaced diagonals

“…. , n − k. However formula ( 16) is valid without this assumption as after simplifications the fractions disappear (see [4]). In our case n − k ≤ k − 1 and the product in ( 16) can be simplified to…”

“…Multidiagonal matrices have a wide range of applications in various field of mathematics and engineering. Among them matrices with equally spaced diagonals have much nicer properties than those with arbitrarily spaced diagonals (see [1,4,5] and their references). Here we study how multidiagonal matrices with equally spaced diagonals behave under multiplication, taking inverse and powers.…”