Finding the rotational matrix that minimizes the sum of squared deviations between two vectors is an important problem in bioinformatics and crystallography. Traditional algorithms involve the inversion or decomposition of a 3 x 3 or 4 x 4 matrix, which can be computationally expensive and numerically unstable in certain cases. Here, we present a simple and robust algorithm to rapidly determine...

The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way. Abstract We argue that the spectrum of the QCD Dirac operator near zero virtuality can be described by random matrix theory. As in the case of classical random matrix ensembles of Dyson we have three different cases: the chiral orthogonal ensemble (chGOE), the chiral unitary ensemble (chGUE) and the chira...

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N ×N matrix, the resulting factorization is a product of O...

In this paper we show that harmonic compact operator-valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis. We also give an example which shows that an operator-valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator-valued function. We us...

in which A is a large sparse symmetric positive definite matrix, λ is an eigenvalue and u is a corresponding eigenvector. The evaluation of one or more smallest eigenpairs has much practical interest for describing the characteristics of physical phenomena. For example, smallest eigenvalues characterize the base frequences of vibrating mechanical structures. Typically, the matrix A is a discret...

‎this paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficients‎. ‎at first‎, ‎the non-self-adjoint spectral problem is derived‎. ‎then its adjoint problem is calculated‎. ‎after that‎, ‎for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined‎. ‎finally the convergence ...

Adjoint symmetry constraints are presented to manipulate binary nonlinearization, and shown to be a slight weaker condition than symmetry constraints in the case of Hamiltonian systems. Applications to the multicomponent AKNS system of nonlinear Schrödinger equations and the multi-wave interaction equations, associated with 3×3 matrix spectral problems, are made for establishing their integrabl...

This paper is a review of results on computational methods of linear algebra over commutative domains. Methods for the following problems are examined: solution of systems of linear equations, computation of determinants, computation of adjoint and inverse matrices, computation of the characteristic polynomial of a matrix.

We show that if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized.

Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...