The linear algebra of the Pascal matrix

Brawer, Robert

doi:10.1016/0024-3795(92)90056-g

Cited by 29 publications

(33 citation statements)

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“…A matrix S is called symplectic if S T JS = J, where J is as in (21). It is easy to check that if H is Hamiltonian, and S is symplectic, then S −1 HS is Hamiltonian.…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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Abstract. Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors:Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.

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“…A matrix S is called symplectic if S T JS = J, where J is as in (21). It is easy to check that if H is Hamiltonian, and S is symplectic, then S −1 HS is Hamiltonian.…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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“…Suppose that ϕ is given by (5). If ω is not a root of unity, then by Theorem 4.1 of [4], the closure of W (C ϕ ) is a disk centered at the origin so that, just as for the other automorphic composition operators, W (C ϕ ) contains 0 in its interior.…”

Section: Conformal Automorphisms and Dilationsmentioning

confidence: 99%

“…The analogous decomposition of the n × n upper left-hand submatrix of T (w) is known (see [5,Theorem 3]), and in fact follows quickly (for w ≤ 1/2) from the discussion above and the fact that C ν leaves invariant the subspace spanned by the monomials {z j : 0 ≤ j ≤ n}. It was, in fact, this result that first led us to the auxiliary mapping ν and the decomposition (25).…”

Section: Sectorial Composition Operatorsmentioning

confidence: 99%

When is zero in the numerical range of a composition operator?

Bourdon

Shapiro

2002

Integr equ oper theory

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Abstract. We work on the Hardy space H2 of the open unit disc U, and consider the numerical ranges of composition operators Cϕ induced by holomorphic self-maps ϕ of U. For maps ϕ that fix a point of U we determine precisely when 0 belongs to the numerical range W of Cϕ, and in the process discover the following dichotomy: either 0 ∈ W or the real part of Cϕ admits a decomposition that reveals it to be strictly positive-definite. In this latter case we characterize those operators that are sectorial. For compact composition operators our work has the following consequences: it yields a complete description of the corner points of the closure of W , and it establishes when W is closed. In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.

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“…In [2] authors discussed the Linear Algebra of the Pascal Matrix, in [14] authors examined the linear algebra of the k-Fibonacci matrix and the symmetric k-Fibonacci matrix. In [12] authors studied on the Pell Matrix.…”

Section: Introductionmentioning

confidence: 99%

On the generalized Perrin and Cordonnier matrices

Sahin

2017

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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C o m m u n . Fa c . S c i. U n iv . A n k . S é r. A 1 M a th . S ta t. Vo lu m e 6 6 , N u m b e r 1 , P a g e s 2 4 2 -2 5 3 (2 0 1 7 ) D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 9 3 IS S N 1 3 0 3 -5 9 9 1 ON THE GENERALIZED PERRIN AND CORDONNIER MATRICES ADEMŞAH · INAbstract. In the present paper, we study the associated polynomials of Perrin and Cordonnier numbers. We de…ne generalized Perrin and Cordonnier matrices using these polynomials. We obtain the inverse of generalized Cordonnier matrices and give some relationships between generalized Perrin and Cordonnier matrices. In addition, we give a factorization of generalized Cordonnier matrices. Finally, we give some determinantal representation of associated polynomials Cordonnier numbers.

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The linear algebra of the Pascal matrix

Cited by 29 publications

References 0 publications

Product Eigenvalue Problems

Product Eigenvalue Problems

When is zero in the numerical range of a composition operator?

On the generalized Perrin and Cordonnier matrices

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