A new decomposition is derived for any complex square matrix. This decomposition is based on the canonical angles between the column space of this matrix and the column space of its conjugate transpose. Some applications of this factorization are given; in particular some matrix partial orderings and the relationship between the canonical angles and various classes of matrices are studied.

The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system’s matrix. A constructive proof of the existence of directional wave-field decomposition with respect to the normal of the boundary is presented. In...

Recent efforts have shown that the reconstruction performance could be improved with optimized sensing matrix according to a given dictionary for a compressed sensing (CS) system. The existed optimizing conditions are mainly used to address the worst-case performance of CS recovery. Considering the quality of a sensing matrix with respect to the mean squared error (MSE) performance of the Oracl...

A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a g...

Matrix decomposition methods are extensively used for Collaborative Filtering in Recommender Systems. This research work investigates the effectiveness of various Matrix decomposition methods for Collaborative Filtering (CF) to predict recommendations. There is a tradeoff between the scalability and quality of predictions; Recommendations made by Singular Value Decomposition (SVD) based algorit...

A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermit...

For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A ∈ G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and ps...

The polar decomposition of an m x n matrix A of full rank, where rn n, can be computed using a quadratically convergent algorithm of Higham SIAMJ. Sci. Statist. Comput., 7 (1986), pp. 1160-1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary A. ...

In this study we examine performance benefits of implementing the QR decomposition in a way that takes advantage of multiple processes or threads. This is done by partitioning the matrix into blocks of a certain number of rows, which is called the blocksize. We examine this algorithm on “tall and skinny” matrices, which are matrices that have a very large number of rows, but comparatively fewer...

The paper presents FPGA based design & implementation of Cholesky Decomposition for matrix calculation to solve least square problem. The Cholesky decomposition has no pivoting but the factorization is stable. It also has an advantage that instead of two matrices, only one matrix multiplied by itself. Hence it requires two times less operation. The Cholesky decomposition has been designed & sim...