This paper is concerned with numerical solutions to general linear matrix equations including the wellknown Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy t...

An efficient computational method is presented for state space analysis of singular systems via Haar wavelets. Singular systems are those in which dynamics are governed by a combination of algebraic and differential equations. The corresponding differential-algebraic matrix equation is converted to a generalized Sylvester matrix equation by using Haar wavelet basis. First, an explicit expressio...

Sylvester double sums versus subresultants given two polynomials A and B first notion symmetric expression of the roots of two polynomials second notion defined through the coefficients polynomials main result of the lecture these two notions are very closely related (idea due to Sylvester [S]) see details and complete proofs in [RS]. 1 Definitions and main result. A and B two finite families o...

If G and H are two cubic graphs, then we write H ≺ G if G admits a proper edge-coloring f with edges of H, such that for each vertex x of G, there is a vertex y of H with f(∂G(x)) = ∂H(y). Let P and S be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if G is a connected bridgeless cubic graph with G...

This paper is concerned with numerical solutions of large scale Sylvester equations AX −XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li andWhite (2002) demonstrated that the so called Cholesky factored ADI method with decent shift parameters can be very effective. In this paper we present a ge...

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...

We consider linear time-invariant (LTI) systems of the following form Σ : { ẋ(t) = Ax(t) + Bu(t), t > 0, x(0) = x, y(t) = Cx(t) + Du(t), t ≥ 0, with stable state matrix A ∈ Rn×n and B ∈ Rn×m, C ∈ Rp×n, D ∈ Rp×m, arising, e.g., from the discretization and linearization of parabolic PDEs. Typically, in practical applications, we have a large state-space dimension n = O(105) and a small input and ...