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

Controllability and controller-observer design for a class of linear time-varying systems

“The final publication is available at Springer via http://dx.doi.org/10.1007/s10852-012-9212-6"In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer

Remarks on band matrices

Elsner L, Redheffer RM. Remarks on band matrices. Numerische Mathematik. 1967;10(2):153-161.In this note we consider band- or tridiagonal-matrices of order k whose elements above, on, and below the diagonal are denoted by b i, a i,c i. In the periodic case, i.e. b i+m=b i etc., we derive for k=nm and k=nm–1 formulas for the characteristic polynomial and the eigenvectors under the assumption that [Pi] m i=1 c ib i>0. In the latter case it is shown that the characteristic polynomial is divisible by the m–1-th minor, as was already observed by Rósa. We also give estimations for the number of real roots and an application to Fibonacci numbers