Bapat, R. B. ; Lal, A. K.
(1994)
*Inequalities for the q-permanent*
Linear Algebra and its Applications, 197-198
.
pp. 397-409.
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(94)90497-9

## Abstract

If A is an n×n matrix and q a complex number, then the q-permanent of A is defined as per_{q} A = Σ _{σ ∈ S}_{n} q^{1(σ)} Πn _{i = 1} a_{iσ}(i) where S_{n} is the symmetric group of degree n and l(σ) denotes the number of inversions of σ [i.e., the number of pairs i,j such that 1≤i<j≤n and σ(/)> σ(/)]. The function is of interest in that it includes both the determinant and the permanent as special cases. It is known that if A is positive semidefinite and if -1≤q≤1, then per_{q}A ≥ 0. We obtain some results for the q-permanent, including Gram's inequality. It has been conjectured by one of the authors that if A is positive definite and not a diagonal matrix, then per_{q}A is strictly increasing in [-1, 1]. We propose some more conjectures.

Item Type: | Article |
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ID Code: | 78323 |

Deposited On: | 19 Jan 2012 06:32 |

Last Modified: | 19 Jan 2012 06:32 |

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