Abstract
We consider new kinds of max and min matrices, $\left[ a_{\max(i,j)}\right] _{i,j\geq1}$ and $\left[ a_{\min(i,j)}\right] _{i,j\geq1},$ as generalizations of the classical max and min matrices. Moreover, their reciprocal analogues for a given sequence $\left\{ a_{n}\right\} $ have been studied. We derive their $LU$ and Cholesky decompositions and their inverse matrices as well as the $LU$-decompositions of their inverses. Some interesting corollaries will be presented.
Highlights
There are many interesting and useful combinatorial matrices defined by a given sequence {an}n≥0
One of them is known as the Hankel matrix and defined as follows:
Let T = {a1, a2, . . . , an} be a finite multiset of real numbers, such that a1 ≤ a2 ≤ · · · ≤ an. They considered the matrices [max(ai, aj)]1≤i,j≤n and [min]1≤i,j≤n defined on the set T
Summary
There are many interesting and useful combinatorial matrices defined by a given sequence {an}n≥0. Fonseca [5] studied general cases of the matrices considered in [13, 20] by defining the matrix [min(ai − b, aj − b)]1≤i,j≤n for a > 0 and a= b. He computed eigenvalues and eigenvectors of this matrix by computing its inverse. They considered the matrices [max(ai, aj)]1≤i,j≤n and [min (ai, aj)]1≤i,j≤n defined on the set T They computed the determinants, inverses, Cholesky decompositions of these matrices and examined positive definiteness of them. We refer to [12]
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