Graded polynomial identities for matrices with the transpose involution

Involution Matrices of Real Quaternions

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

Invariants and polynomial identities for higher rank matrices

We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct discriminants and the determinant as the discriminant of order d, where d is the dimension of the matrix. The characteristic polynomials and the Cayley–Hamilton th...

Partial Transpose of Permutation Matrices

The main purpose of this paper is to look at the notion of partial transpose from the combinatorial side. In this perspective, we solve some basic enumeration problems involving partial transpose of permutation matrices. Specifically, we count the number of permutations matrices which are invariant under partial transpose. We count the number of permutation matrices which are still permutation ...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Polynomial identities for partitions

For any partition λ of an integer n , we write λ =< 11, 22, . . . , nn > where mi(λ) is the number of parts equal to i . We denote by r(λ) the number of parts of λ (i.e. r(λ) = ∑n i=1mi(λ) ). Recall that the notation λ ` n means that λ is a partition of n . For 1 ≤ k ≤ N , let ek be the k-th elementary symmetric function in the variables x1, . . . , xN , let hk be the sum of all monomials of to...