Completely positive matrices by Abraham Berman

By Abraham Berman

A true matrix is confident semidefinite if it may be decomposed as A=BB¢. In a few functions the matrix B should be elementwise nonnegative. If one of these matrix exists, A is named thoroughly optimistic. The smallest variety of columns of a nonnegative matrix B such that A=BB¢ is called the cp-rank of A.

This priceless publication makes a speciality of priceless stipulations and enough stipulations for entire positivity, in addition to bounds for the cp-rank. The equipment are combinatorial, geometric and algebraic. the necessary history on nonnegative matrices, cones, graphs and Schur enhances is printed.

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If the WLS estimator is properly a p p l i e d it will still b e consistent e v e n w h e n the disturbing n o i s e is other than normally distributed, but it is n o longer asymptotically efficient, a n d c o n s e q u e n t i y the uncertainty o n the estimates will increase c o m p a r e d with that o f the c o r r e s p o n d i n g MLE. This p h e n o m e n o n is illustrated in the following e x a m p l e . Consider a set of m e a s u r e m e n t s disturbed b y independently L a p l a c e distributed n o i s e n^.

48) a n d taking the logarithm g i v e s In L ( z J P ) = l n ( ^ . 51) T h e first part of this e x p r e s s i o n is a constant with respect to the parameters P. 52) This is n o m o r e than a quadratic form with a weighting matrix R ^ T h e estimator o b t a i n e d by the minimization o f Κ is called the Markov estimator. If the weighting matrix is not the covariance matrix but an arbitrary positive definite matrix, the estimator is called a weighted least squares estimator. A GENERAL INTRODUCnON TO PARAMETER ESTIMATON 29 It is o b v i o u s that if the n o i s e o n the m e a s u r e m e n t s is additive a n d normally distributed with z e r o m e a n a n d covariance matrix R, then the M a r k o v estimator a n d the MLE are identical.

22) 40 ΙΟΕΝΉΡΙΟΑΉΟΝ OF UNEAR SYSTEMS T h e minimization o f Κ with respect to Ρ results in a set of linear e q u a t i o n s which must b e s o l v e d . 23) This set o f equations is called the normal equations. 23) is preferred for calculating the estimates, but for theoretical analysis e x p r e s ­ sion (2,24) is m o r e suitable. 25) = P+0 if£{n ) = 0 y T h e b i a s is zero if the m e a n value o f the n o i s e is zero. 26) with Κ=£{η^η^'} the covariance matrix of the noise perturbations.

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