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1

Binary Symmetric Matrix Inversion Through Local Complementation

100%

Brijder R. , Hoogeboom H. J.

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tom Vol. 116, nr 1-4

15-23

EN

We consider the Schur complement operation for symmetric matrices over GF(2), which we identify with graphs through the adjacency matrix representation. It is known that Schur complementation for such a matrix (i.e., for a graph) can be decomposed into a sequence of two types of elementary Schur complement operations: (1) local complementation on a looped vertex followed by deletion of that vertex and (2) edge complementation on an edge without looped vertices followed by deletion of that edge. We characterize the symmetricmatrices over GF(2) that can be transformed into the empty matrix using only operations of (1). As a consequence, we find that these matrices can be inverted using local complementation. The result is applied to the theory of gene assembly in ciliates.

2

Some properties of the spectral radius of a set of matrices

80%

Czornik A. , Jurgas P.

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nr 2

183-188

EN

In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.

3

Immanant Conversion on Symmetric Matrices

60%

Purificação Coelho M. , Duffner M. A. , Guterman A. E.

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nr 1

EN

Letr Σn(C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C) -> Σn (C) satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB) = dχ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С).

4

Some properties of the spectral radius of a set of matrices

60%

Czornik A. , Jurgaś P.

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tom Vol. 16, no 2

183-188

EN

In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.