The Concentration Ellipsoid of a Random Vector Revisited

1991 ◽  
Vol 7 (3) ◽  
pp. 397-403 ◽  
Author(s):  
Kenneth Nordström
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Crucial Role ◽  
Type Inequality ◽  
Estimation Theory ◽  
Linear Estimation ◽  
Nonnegative Definite

Alternative definitions of the concentration ellipsoid of a random vector are surveyed, and an extension of the concentration ellipsoid of Darmois is suggested as being the most convenient and natural definition. The advantage of the proposed definition in providing substantially simplified proofs of results in (linear) estimation theory is discussed, and is illustrated by new and short proofs of two key results. A not-so-well-known, but elementary, extremal representation of a nonnegative definite quadratic form, together with the corresponding Cauchy-Schwarẓ-type inequality, is seen to play a crucial role in these proofs.

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

Orlicz type inequalities with powers

2015 ◽  
Vol 26 (13) ◽  
pp. 1550111
Author(s):  
Cholryong Kang ◽  
Songil Ri
Keyword(s):  
Quadratic Form ◽  
Dirichlet Form ◽  
Sobolev Inequality ◽  
Type Inequality

Basing on the Orlicz–Sobolev inequality for a given Dirichlet form, the same type inequality is established for the quadratic form associated to the generator with a power. Some examples are provided to show the optimality of the main result.

scholarly journals Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

2019 ◽  
Vol 125 (2) ◽  
pp. 239-269 ◽  
Author(s):  
Ari Laptev ◽  
Michael Ruzhansky ◽  
Nurgissa Yessirkegenov
Keyword(s):  
Magnetic Field ◽  
Quadratic Form ◽  
Type Inequality ◽  
Hardy Inequalities ◽  
Landau Hamiltonian ◽  
Grushin Operator ◽  
Aharonov Bohm ◽  
The Magnetic Field

In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian type magnetic field. Moreover, we obtain a Poincaré type inequality and inequalities with more general families of weights. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of the variables. The corresponding uncertainty type principles are also obtained.

Linear estimation theory applied to the reconstruction of a 3-D vector current distribution

1990 ◽  
Vol 29 (5) ◽  
pp. 658 ◽  
Author(s):  
Warren E. Smith ◽  
William J. Dallas ◽  
Walter H. Kullmann ◽  
Heidi A. Schlitt
Keyword(s):  
Current Distribution ◽  
Vector Current ◽  
Estimation Theory ◽  
Linear Estimation
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