23 research outputs found
Full-Rank Factorization and Moore-Penrose’s Inverse
C. MacDuffee apparently was the first to point out, in private communications, that a full-rank factorization of a matrix A leads to an explicit formula for its Moore-Penrose’s inverse A+. Here we apply this idea of MacDuffee and the Singular Value Decomposition to construct A+
Computing generalized inverses using LU factorization of matrix product
An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and
the Moore-Penrose inverse of a given rational matrix A is established. Classes
A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R*
and T*(AT*)+, where R and T are rational matrices with appropriate dimensions
and corresponding rank. The proposed algorithm is based on these general
representations and the Cholesky factorization of symmetric positive matrices.
The algorithm is implemented in programming languages MATHEMATICA and DELPHI,
and illustrated via examples. Numerical results of the algorithm, corresponding
to the Moore-Penrose inverse, are compared with corresponding results obtained
by several known methods for computing the Moore-Penrose inverse
INVERS MOORE-PENROSE SEBAGAI REPRESENTASI MATRIKS PROYEKSI ORTHOGONAL
The inverse of matrix is one of the important properties of matrix. This properies, especially singular matrix, has been developed by Moore and continued by Penrose. Then, this inverse called Moore-Penrose inverse. The Moore-Penrose invers criteria can represent a projection on a vector space V along W with V and W are orthogonal to each other or can written with W=V^⊥ which is called orthogonal projection matrix on V. This research will present lemmas and theorems related to the Moore-Penrose invers construction of the multiplication matrix. Then, a square matrix is an orthogonal projection matrix on a vector space V if and only if it satisfies two conditions, that are idempotent and symmetric. These two properties are satisfied by matrices I-A^+ A and I-AA^+ which respectively are orthogonal projection matrices on Ker(A) and Ker(A^' ). As a result, the Moore-Penrose inverse A^+ can be constructed from a square matrix A which is an multiplication of several matrices and fulfills certain properties
Dealing with Interference in Distributed Large-scale MIMO Systems: A Statistical Approach
This paper considers the problem of interference control through the use of
second-order statistics in massive MIMO multi-cell networks. We consider both
the cases of co-located massive arrays and large-scale distributed antenna
settings. We are interested in characterizing the low-rankness of users'
channel covariance matrices, as such a property can be exploited towards
improved channel estimation (so-called pilot decontamination) as well as
interference rejection via spatial filtering. In previous work, it was shown
that massive MIMO channel covariance matrices exhibit a useful finite rank
property that can be modeled via the angular spread of multipath at a MIMO
uniform linear array. This paper extends this result to more general settings
including certain non-uniform arrays, and more surprisingly, to two dimensional
distributed large scale arrays. In particular our model exhibits the dependence
of the signal subspace's richness on the scattering radius around the user
terminal, through a closed form expression. The applications of the
low-rankness covariance property to channel estimation's denoising and
low-complexity interference filtering are highlighted.Comment: 12 pages, 11 figures, to appear in IEEE Journal of Selected Topics in
Signal Processin
Computation of generalized inverses by using the LDL∗ decomposition
AbstractAn efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse
Fast solving of Weighted Pairing Least-Squares systems
This paper presents a generalization of the "weighted least-squares" (WLS),
named "weighted pairing least-squares" (WPLS), which uses a rectangular weight
matrix and is suitable for data alignment problems. Two fast solving methods,
suitable for solving full rank systems as well as rank deficient systems, are
studied. Computational experiments clearly show that the best method, in terms
of speed, accuracy, and numerical stability, is based on a special {1, 2,
3}-inverse, whose computation reduces to a very simple generalization of the
usual "Cholesky factorization-backward substitution" method for solving linear
systems