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1. An identity between the determinant and the permanent of Hessenberg-type matrices

Creator:
Da Fonseca, Carlos Martins
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
determinant, permanent, Hessenberg matrices;, graphs, and trees
Language:
English
Description:
In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson involving Hessenberg matrices is consequently generalized.
Rights:
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2. Determinants of matrices associated with incidence functions on posets

Creator:
Hong, Shaofang and Sun, Qi
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
meet-closed set, greatest-type lower, incidence function, determinant, and nonsingularity
Language:
English
Description:
Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $[f(x_i\wedge x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $[f(x_i\vee x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $[f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices $f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$ to be nonsingular. Finally, we give some number-theoretic applications.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Integer matrices related to Liouville's function

Creator:
Oon, Shea-Ming
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Liouville's function, determinant, and LU decomposition
Language:
English
Description:
In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion may be extended to that of Riemann hypothesis.
Rights:
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4. Nested matrices and inverse M-matrices

Creator:
Stuart, Jeffrey L
Format:
print, bez média, and svazek
Type:
model:article and TEXT
Subject:
matematika, mathematics, nested matrix, tridiagonal matrix, inverse M-matrix, principal minor, determinant, QR-factorization, 13, and 51
Language:
English
Description:
Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the LU- and QR-factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse M-matrices with symmetric, irreducible, tridiagonal inverses., Jeffrey L. Stuart., and Obsahuje seznam literatury
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public