Abstract
We 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.
Highlights
Let Mn (C) denote the space of all n-square matrices over the complex field C and let Sn be the symmetric group of degree n
In [3], Gibson proved that if A is an n-square (0, 1)-matrix, and if the permanent of A can be converted to a determinant by affixing ± signs to the elements of the matrix, A has at most
We extend Fonseca’s result [8] by presenting an explicit characterization of convertible Hessenberg-type matrices and corresponding subspaces
Summary
Let Mn (C) denote the space of all n-square matrices over the complex field C and let Sn be the symmetric group of degree n. In [3], Gibson proved that if A is an n-square (0, 1)-matrix, and if the permanent of A can be converted to a determinant by affixing ± signs to the elements of the matrix, A has at most. We consider mostly n-square (0, 1)-matrices with the maximum number of positive entries Ωn. For these cases, we present a procedure to determine whether or not a given matrix is convertible. Compared with previously available algorithms, this method does not rely on the associated bipartite graph, and is more efficient This result is presented, where we introduce a new concept: the imprint. We conclude that convertible matrices can be reduced to a basic set
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