The spectral decomposition of normal linear (bounded) operators and the polar decomposition of arbitrary linear (bounded) operators on Hilbert spaces have been interesting and technically useful results in operator theory [3, 9, 13, 20]. The development of the concept of von Neumann algebras on Hilbert spaces has shown that both these decompositions are always possible inside of the appropriate...

The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divideand-conquer algorithms, which recursively...

In this paper we propose a new algorithm for the joint eigenvalue decomposition of a set of real non-defective matrices. Our approach resorts to a Jacobi-like procedure based on polar matrix decomposition. We introduce a new criterion in this context for the optimization of the hyperbolic matrices, giving birth to an original algorithm called JDTM. This algorithm is described in detail and a co...

The polar decomposition A = UpH finds many uses in applications, and it is a fundamental tool for computing the symmetric eigenvalue decomposition and the singular value decomposition via a spectral divide-and-conquer process. Conventional algorithms for these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with a...

1 I n t r o d u c t i o n . This work appl ies some l inear a lgebra ideas in an o rd ina ry differential equa t ion (ODE) context . We begin by summar i s ing the a p p r o p r i a t e l inear a lgebra , and then we in t roduce the O D E problem. *Received July 1995. Revised July 1996. tThis work was supported by Engineering and Physical Sciences Research Council grants GR/H94634 and GR/KS0228...

Let L(λ) = Inλ m + Am−1λm−1 + · · ·+ A1λ + A0 be an n× n monic matrix polynomial, and let CL be the corresponding block companion matrix. In this note, we extend a known result on scalar polynomials to obtain a formula for the polar decomposition of CL when the matrices A0 and Pm−1 j=1 AjA ∗ j are nonsingular.

An explicit formula for the polar decomposition of an n n nonsingular companion matrix is derived. The proof involves the largest and smallest singular values of the companion matrix.

Local decompositions of a Dirac spinor into ‘charged’ and ‘real’ pieces ψ(x) = M(x)χ(x) are considered. χ(x) is a Majorana spinor, andM(x) a suitable Dirac-algebra valued field. Specific examples of the decomposition in 2 + 1 dimensions are developed, along with kinematical implications, and constraints on the component fields within M(x) sufficient to encompass the correct degree of freedom co...

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. ...