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The Ramanujan Journal - Answering a question of Erdös, Komlós proved in 1968 that almost all $$ntimes n$$ Bernoulli matrices are nonsingular as $$nrightarrow infty $$ . In this paper,...  相似文献   

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Let A, B, P, Q be n-square nonsingular complex matrices, and for 1 ?m <n let U and V be n×m complex matrices. For n?3 necessary and sufficient conditions are given for the inequality
det(U1AV)det(V1BU)?det(U1PU)det(V1QV)
to hold for all U and V.  相似文献   

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Let Δ(α + β) = |Hλ2?r+1| where Hr is the complete symmetric function in (α1 + β1), (α2 + β2), …, (αn + βn). It is proved that Δ(α + β) ? Δ(α) + Δ(β). This inequality is generalised for certain symmetric functions defined by Littlewood. Let Ω(α + β) = |Qλ2?r+1staggered(α + β) (t, k1, k2, …, km)|. Then we prove that Ω(α + β) ? Ω(α) + Ω(β). Here λ1, λ2, λ3, …, λn is a partition such that λn > λn?1 > ··· > λ2 > λ1.  相似文献   

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A generalization of the Maillet determinant and the Demyanenko matrix is given.  相似文献   

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A new algorithm for treating numerically a pentadiagonal matrix is given. As an application the evaluation of the eigenvalues is performed. Also the stability of the solution of some difference equations is examined.  相似文献   

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In this note we prove that there is no linear mapping T on the space of n-square symmetric matrices over any subfield of real field such that the determinant of A is equal to the permanent of T(A) for all symmetric matrices A if n≥3.  相似文献   

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Hadamard's determinant theorem is used to obtain an upper bound for the modulus of the determinant of the sum of two normal matrices in terms of their eigenvalues. This bound is compared with another given by M. E. Miranda.  相似文献   

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Let M be a von Neumann algebra with a faithful semifinite trace τ. For τ-compact elements x?M, Grothendieck introduced in “Séminaire Bourbaki,” 1955 a function Δ1 + x generalizing the Fuglede-Kadison determinant, and conjectured that
Δ1 + ¦x + y¦ ? Δ1 + ¦x¦Δ1+ ¦y¦
. In this paper, the inequality is proved. As a corollary, a direct proof of the Clarkson inequalities is obtained.  相似文献   

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We prove a formula expressing a generaln byn Toeplitz determinant as a Fredholm determinant of an operator 1 –K acting onl 2 (n,n+1,...), where the kernelK admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see [14], and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.  相似文献   

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When the directed graph of an n-by-n matrix A does not contain a Hamilton cycle, we exhibit a formula for det A in terms of sums of products of proper principal minors of A. The set of minors involved depends upon the zero/nonzero pattern of A.  相似文献   

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Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this method is much simpler.  相似文献   

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One considers the asymptotic behavior of the Toeplitz determinant for a nonnegative function. One proves that under certain conditions on the function the determinant admits the asymptotic representation where Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 22–31, 1980.In conclusion I express my gratitude to V. N. Solev for the formulation of the problem and for his help in the preparation of the paper.  相似文献   

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Denote byH n the set ofn byn, positive definite hermitian matrices. Hadamard proved thath(A)≧det(A) for allAH n, whereh(A) is the product of the main diagonal elements ofA. Subsequently, M. Marcus showed that per(A)h(A) for allAH n. This article contains a result for all generalized matrix functions from which it follows thath(A)≧(per(A1/n )) n ,AH n.  相似文献   

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