Bapat, R. B.
(1991)
*An interlacing theorem for tridiagonal matrices*
Linear Algebra and its Applications, 150
.
pp. 331-340.
ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0024-3795(91)90178-Y

## Abstract

If A is an n × n matrix and if S ⊂{1,...,n}, then let A(S) denote the principal submatrix of A formed by rows and columns in S. If A, B are n × n matrices, then let η(A, B) = Σ_{s}det A(S) det B(S^{t}) where the summation is over all subsets of {1,...,n}, where S' denotes the complement of S, and where, by convention det A(φ) = det B(φ) = 1. It has been conjectured that if A is positive definite and B hermitian, then the polynomial η(λA, - B) has only real roots. We prove this conjecture if n ≤ 3, and also for any n under the additional assumption that both A, B are tridiagonal. We derive some consequences, including a generalization of a majorization result of Schur for tridiagonal matrices.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

ID Code: | 78324 |

Deposited On: | 19 Jan 2012 06:31 |

Last Modified: | 19 Jan 2012 06:31 |

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