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- Equality of immanantal decomposable tensors, IIPublication . Cruz, Henrique F. Da; Silva, J. A. Dias daWe state a necessary and sufficient condition for equality of nonzero decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of the symmetric group of degree m.
- Maximal doubly stochastic matrix centralizersPublication . Cruz, Henrique F. da; Dolinar, Gregor; Fernandes, Rosário; Kuzma, BojanWe describe doubly stochastic matrices with maximal central-izers.
- Convertible subspaces that arise from different numberings of the vertices of a graphPublication . Cruz, Henrique F. Da; Inácio, Ilda; Serôdio, RogérioIn this paper, we describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. These subspaces can arise from different numberings of the vertices of a graph. With this numbering process, we obtain some well-known sequences of integers. For instance, in the case of a path of length n, we prove that the number of these subspaces is the (n + 1)th Fibonacci number.
- Convertible Subspaces of Hessenberg-Type MatricesPublication . Cruz, Henrique F. Da; Rodrigues, Ilda Inácio; Serôdio, Rogério; Simões, A. M.; Velhinho, JoseWe describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set.
- The number of P-vertices in a matrix with maximum nullityPublication . Fernandes, Rosário; Cruz, Henrique F. DaLet T be a tree with n≥2 vertices. Set S(T) for the set of all real symmetric matrices whose graph is T. Let A∈S(T) and i∈{1,…,n} . We denote by A(i) the principal submatrix of A obtained after deleting the row and column i. We set mA(i)(0)=mA(0)+1, we say that i is a P-vertex of A. As usual, M(T) denotes the maximum nullity occurring of B∈S(T). In this paper we determine an upper bound and a lower bound for the number of P-vertices in a matrix A∈S(T)with nullity M(T). We also prove that if the integer b is between these two bounds, then there is a matrix E∈S(T) with b P-vertices and maximum nullity.