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Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley, Seok-Zun Song, Young Bae Jun (2014)

Czechoslovak Mathematical Journal

Let A be a Boolean { 0 , 1 } matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A . A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O , to itself. A mapping strongly preserves a set, S , if it maps the set S into the set S and the complement of...

Linear preserver of n × 1 Ferrers vectors

Leila Fazlpar, Ali Armandnejad (2023)

Czechoslovak Mathematical Journal

Let A = [ a i j ] m × n be an m × n matrix of zeros and ones. The matrix A is said to be a Ferrers matrix if it has decreasing row sums and it is row and column dense with nonzero ( 1 , 1 ) -entry. We characterize all linear maps perserving the set of n × 1 Ferrers vectors over the binary Boolean semiring and over the Boolean ring ℤ 2 . Also, we have achieved the number of these linear maps in each case.

Linear preservers of row-dense matrices

Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall (2016)

Czechoslovak Mathematical Journal

Let 𝐌 m , n be the set of all m × n real matrices. A matrix A ∈ 𝐌 m , n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T : 𝐌 m , n → 𝐌 m , n that preserve or strongly preserve row-dense matrices, i.e., T ( A ) is row-dense whenever A is row-dense or T ( A ) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ 𝐌 n , m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear...

Linear Transformations of Euclidean Topological Spaces

Karol Pąk (2011)

Formalized Mathematics

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.

Linear transforms supporting circular convolution over a commutative ring with identity

Mohamed Mounir Nessibi (1995)

Commentationes Mathematicae Universitatis Carolinae

We consider a commutative ring R with identity and a positive integer N . We characterize all the 3-tuples ( L 1 , L 2 , L 3 ) of linear transforms over R N , having the “circular convolution” property, i.eṡuch that x * y = L 3 ( L 1 ( x ) ⊗ L 2 ( y ) ) for all x , y ∈ R N .

Linearization of Poisson actions and singular values of matrix products

Anton Alekseev, Eckhard Meinrenken, Chris Woodward (2001)

Annales de l’institut Fourier

We prove that the linearization functor from the category of Hamiltonian K -actions with group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian K - actions, preserves products up to symplectic isomorphism. As an application, we give a new proof of the Thompson conjecture on singular values of matrix products and extend this result to the case of real matrices. We give a formula for the Liouville volume of these spaces and obtain from it a hyperbolic version of the Duflo...

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