The relatively universal cover of the natural embedding of the long root geometry for the group SL ( n + 1 , 𝕂 )

This note is a comment to [1], where it was proved that, over a field of odd characteristic, the number of square matrices of order n, n ? 3, with nonzero permanent is greater than the number of square matrices of the same order with nonzero determinant.

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers zn, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f(z) as follows: [f(z + a) − f(z)]/a respectively [f(qz) − f(z)]/[(q − 1) z]. These representations are exact—in a sense explained in the paper—when the function f(z) is a polynomial in z of degree less than N. This formalism allows to transform difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique, several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers zn, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character.

In 1958, Gerstenhaber showed that if is a subspace of the vector space of the square matrices of order n over some field consisting of nilpotent matrices only (to be called a nilspace) and if the underlying field is sufficiently large, then the maximal dimension of is . This dimension is attained if and only if the linear space is similar to the space of all strictly upper-triangular matrices. In this paper, we study maximal spaces of nilpotent square matrices of order n. As a striking extension of the Gerstenhaber’s result, we prove that a maximal nilspace (with the underlying field being sufficiently large) is similar to a (subspace of) all strictly upper-triangular matrices if and only if it contains a nilpotent J of maximal possible rank and its square . We give a twisted but elementary proof of this fact.

Linear spaces of symmetric nilpotent matrices

On “P” property and the column-W property

We consider the set of all equitable matrices of order n for a given partitionof n. We show that this set is a subring of the ring of square matrices of order n.Moreover, we prove that this set is epimorphic to the ring of square matrices of order k,where k is the number of parts in the given partition. In particular, we prove that thesum, the product and the inverse of equitable matrices are also equitable

We consider the algebra of square matrices of order N as graded q-differential algebra, where q is a primitive Nth root of unity. We study the differential calculus of this algebra determined by a differential d, which satisfies the graded q-Leibniz rule and dN = 0. A graded q-differential algebra can be viewed as a generalization of graded differential algebra, and we consider the higher degree elements of graded q-differential matrix algebra as analogs of differential forms. We use two matrices x,ξ, which generate the algebra of square matrices of order N in order to express these analogs of differential forms in terms of "coordinate" x and its differentials dx,d2x,...,dN−1x. We study an analog of connection and calculate its curvature.

This work arose from the following question asked by a student on an electronics technical course in 2012: “Is the Sar-rus rule only valid for determinants of square matrices of order three?” . Responding to this challenging question and questions such as Thomas Muir's and R. Osborn's objection, we give the reader unprecedented material that presents in a didactic way (and with demonstration) a generalization for Sarrus' method applying it to matrices of arbitrary order. We will call the generalization method the generalized sarrus method or the STAF method (acronym for Sarrus-Trovão-Almeida-Ferreira).

Dedicated to the memory of my teacher Prof. VOJTMCH JARNIKLet M n be the space of square matrices of order n, R -the real line, R + -the positive halfline (closed), R" -the negative halfline, A : R~ -> M n , B : R~ -> M n locally integrable.For y e R n denote by |>;| the Euclidean norm of y and for C e M n put \C\ = sup \Cy\.

This paper deals with the evaluation of preference consistency in decision making, assuming that decision makers express their preferences by means of pairwise comparisons in the set of alternatives. Preferences can be expressed using one of the various known representations, such as fuzzy preference relations or multiplicative pairwise comparison matrices. A geometrical characterization of inconsistency evaluation is proposed by considering a pairwise comparison matrix as a point in the vector space of square matrices of order n and by using different metrics to measure deviation of this matrix from full consistency. An inconsistency index is defined as the minimum distance of a pairwise comparison matrix from a consistent one, according to a fixed metric. Consequently, to each choice of a particular metric corresponds an inconsistency index. Geometrical properties of the subset of consistent matrices are investigated.

Spatial decomposition of functionally commutative matrices

Consider the vector space of square matrices of order r and the corresponding projective space ℙ = ℙ r 2−1. The points of ℙ are in a one-to-one correspondence with the square matrices modulo multiplication by a nonzero constant. Consider the Segre subvariety corresponding to the matrices of rank one and a filtration where, for 1 ≤ i ≤ r, X i denotes the i-th join of X with itself. We recall that by definition X i (also called the (i − 1)-st secant variety of X) is the closure of the subvariety of ℙ swept out by the linear spans of general collections of i points of X. Thus in our case X i corresponds to the cone of matrices whose rank does not exceed i. In other words, if M is a matrix and z M ∈ ℙ is the corresponding point, then rk M = rk X z M , where for z ∈ ℙ .

In this chapter we collect the definitions and some of the most important properties of square matrices of order 2.

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.

Primitive Elements in the Finite Field of Square Matrices of Order 2 for Cryptographic Applications