Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

A Chain Rule for Matrix Functions and Applications

Abstract: Let $f$ be a not necessarily analytic function and let $A(t)$ be a family of $n \times n$ matrices depending on the parameter $t$. Conditions for the existence of the first and higher derivatives of $f(A(t))$ are presented together with formulae that represent these derivatives as a submatrix of $f(B)$, where $B$ is a larger block Toeplitz matrix. This block matrix representation of the first derivative is shown to be useful in the context of condition estimation for matrix functions. The results presented here are slightly stronger than those in the literature and are proved in a considerably simpler way.

Network design problems for controlling virus spread

Abstract: The spread of viruses in human populations (e.g., SARS) or computer networks is closely related to the network's topological structure. In this paper, we study the problem of allocating limited control resources (e.g., quarantine or recovery resources) in these networks to maximize the speed at which the virus is eliminated, by exploiting the topological structure. This problem can be abstracted to that of designing diagonal K or D to minimize the dominant eigenvalue of one of the system matrices KG, D + KG or D + G under constraints on K and D (where G is a square matrix that captures the network topology). We give explicit solutions to these problems, using eigenvalue sensitivity ideas together with constrained optimization methods employing Lagrange multipliers. Finally, we show that this decentralized control approach can provide significant advantage over a homogeneous control strategy, using a model for SARS transmission in Hong Kong derived from real data.

A Comparison of Representations of General Spatial Screw Displacement

Abstract: This paper presents a comparison of eight methods of representing a general spatial rigid-body screw displacement. A distinction is made between point transformations and line transformations. The following three representations involving point transformations are considered: the real 4 × 4 matrix; the complex 4 × 4 matrix; and the real 6 × 6 matrix. These utilise homogeneous coordinates. The remaining five representations considered here are concerned with line transformations and are: the dual orthogonal 3 × 3 matrix; the dual special unitary 2 × 2 matrix; the dual Pauli spin matrices; the dual unit quaternion; and the dual special unitary 3 × 3 matrix. These make use of dual-number techniques. The most commonly used of these representations is the 4 × 4 real matrix and the conclusion reached is that this is certainly the best of the point transformations. The dual unit quaternion is similarly found to be the best of the line transformations.

Smoothing-Norm Preconditioning for Regularizing Minimum-Residual Methods

Abstract: When GMRES (or a similar minimum-residual algorithm such as RRGMRES, MINRES, or MR-II) is applied to a discrete ill-posed problem with a square matrix, in some cases the iterates can be considered as regularized solutions. We show how to precondition these methods in such a way that the iterations take into account a smoothing norm for the solution. This technique is well established for CGLS, but it does not immediately carry over to minimum-residual methods when the smoothing norm is a seminorm or a Sobolev norm. We develop a new technique which works for any smoothing norm of the form $\|L\,x\|_2$ and which preserves symmetry if the coefficient matrix is symmetric. We also discuss the efficient implementation of our preconditioning technique, and we demonstrate its performance with numerical examples in one and two dimensions.