On positive eigenvectors of positive infinite matrices

Kato, Tosio

doi:10.1002/cpa.3160110407

Cited by 5 publications

(3 citation statements)

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“…For example, using the kernel (1), with ℓ = 0, we get the (generalized) Hilbert matrix. Matrices of this type have been explored, for instance, in [7]. But as emphasized in the introduction of the cited paper, it appears that there may be inconveniences in applying to matrices some methods originally invented for integral operators.…”

Section: Introductionmentioning

confidence: 99%

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Kalvoda

Šťovı́ček

2015

Linear and Multilinear Algebra

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A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced with the entriessupposing a, b, c are positive and a < b + c, b < a + c, c ≤ a + b. The famous Hilbert matrix is included as a particular case. The direct sum B(a, b, c) ⊕ B(a + 1, b + 1, c) is shown to commute with a discrete analog of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, T (a, b, c), commuting with B(a, b, c). The orthogonal polynomials associated with T (a, b, c) turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping U diagonalizing T (a, b, c) can be constructed explicitly. At the same time, U diagonalizes B(a, b, c) and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval [0, M (a, b, c)] where M (a, b, c) is known explicitly.If the assumption c ≤ a + b is relaxed while the remaining inequalities on a, b, c are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold M (a, b, c). Again, all eigenvalues and eigenvectors are described explicitly.

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Section: Introductionmentioning

confidence: 99%

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Kalvoda

Šťovı́ček

2015

Linear and Multilinear Algebra

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“…In [2] it was shown that A -B is compact and 1 = \\B\\ e = \\A\\ e < ||A||, and in particular A has eigenvalues, thus distinguishing its spectral properties from those of the Hilbert matrix [1]. In fact, the relationship between A and B turns out to be a particular case of the general form of matrices with principal homogeneous part studied by T. Kato [4]. For this and other reasons which we hope will be clear in this note, B turns out to be an interesting matrix in its own right.…”

Section: U^omentioning

confidence: 99%

On the Spectrum of the Bergman-Hilbert Matrix II

Davis¹,

Ghatage²

1990

Can. math. bull.

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“…On the other hand, it may happen that there exists a vector x not in Hilbert space for which Ax =nx. This problem has been considered in particular by Kato [9], [10] and Rosenblum [17]. The Hilbert matrix A = ((/+y)~1) sat isfi es '4 > 0 and is bounded; in fact H = n (see [6], Chapter IX) and, moreover, [i is not in the point spectrum of A ( [12], [18]).…”

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confidence: 99%

Positive matrices and eigenvectors

Putnam

1963

Proceedings of the Glasgow Mathematical Association

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For i, j = 1, 2, …, let aij be real. A matrix A = (aij) will be called positive (A>0) or non-negative (A≧0) according as, for all i and j, aij>0 or aij≧0 respectively. Correspondingly, a real vector x = (x1, x2, …) will be called positive (x>0) or non-negative (x≧0) according as, for all i, xi>0 or x≧0. A matrix A is said to be bounded if ∥ Ax ∥ ≦M ∥ x ∥ holds for some constant M, 0 ≦ M < ∞, and all x in the Hilbert space H of real vectors x = (x1, x2, …) satisfying . The least such constant M is denoted by ∥ A ∥. If x and y belong to H, then (x, y) will denote as usual the scalar product Σxiyi. Whether or not x is in H, or A is bounded, y = Ax will be considered as defined bywhenever each of the series of (1) is convergent.

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On positive eigenvectors of positive infinite matrices

Cited by 5 publications

References 7 publications

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

On the Spectrum of the Bergman-Hilbert Matrix II

Positive matrices and eigenvectors

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