Bapat, R. B.
(1989)
*Mixed discriminants of positive semidefinite matrices*
Linear Algebra and its Applications, 126
.
pp. 107-124.
ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002437...

Related URL: http://dx.doi.org/10.1016/0024-3795(89)90009-8

## Abstract

If A^{k}=(a^{k}_{ij}), k= 1,2,…,n, are n-by-n matrices, then their mixed discriminant D(A^{1},…,A^{n}) is given by D(A^{1}.....A^{n}=1/n! Σ_{σ∈}S_{a}| (α_{ij}^{α(j)})| where S_{n} is the symmetric group of degree n and where |·| denotes determinant. We give certain alternative ways of defining the mixed discriminant and state some basic properties. It is pointed out that a Ryser-type formula for the mixed discriminant exists in the literature, and a simpler proof is given for it. It is shown that the mixed discriminant can be expressed as an inner product. A generalization of Konig's theorem on 0-1 matrices is proved. The following set D_{n}, which includes the set of n-by-n doubly stochastic matrices, is defined and studied: D_{n}={(A^{1}.....,A^{n}):A^{i} is a-by-n, positive semidefinite with trace 1, i=1,2,....n; Σ^{n}_{i=1}A^{i}=1}.

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

ID Code: | 1405 |

Deposited On: | 05 Oct 2010 12:36 |

Last Modified: | 13 May 2011 08:13 |

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