Masaya Matsuura

On the Craig–Sakamoto theorem and Olkin’s determinantal result

Abstract Let A and B be any n×n real symmetric matrices. The following fact is well known: If |In−αA−βB|=|In−αA||In−βB| for any α, β∈ R , then AB=0. There exist various proofs. In this paper, we refine Olkin’s method [Linear Algebra Appl. 264 (1997) 217]. Furthermore, his determinantal result is generalized.

On a New Fluctuation–Dissipation Theorem for Degenerate Stationary Flows

The theory of KM2O-Langevin equations for stochastic processes (or more generally, flows in inner product spaces) have been developed in view of applications to time series analysis (e.g., Okabe and Nakano, 1991; Okabe, 1999, 2000; Okabe and Matsuura, 2000). In Klimek et al. (2002) and Matsuura and Okabe (2001, 2003), we have investigated degenerate flows, which is important in the analysis of time series obtained from deterministic dynamical systems. As a continuation, we shall in this paper derive an efficient algorithm by which the minimum norm coefficients of KM2O-Langevin equations are explicitly obtained in degenerate cases. The obtained results have close relations to the calculations of conditional expectations such as nonlinear predictors of stochastic processes (Matsuura and Okabe, 2001). The method has also potential applications to financial mathematics.

A generalization of Moore-Penrose biorthogonal systems

In this paper, the notion of Moore-Penrose biorthogonal systems is generalized. In (Fiedler, Moore-Penrose biorthogonal systems in Euclidean spaces, Lin. Alg. Appl. 362 (2003), pp. 137-143), transformations of generating systems of Euclidean spaces are examined in connection with the Moore-Penrose inverses of their Gram matrices. In this paper, g-inverses are used instead of the Moore-Penrose inverses and, in particular, the details of transformations derived from reflexive g- inverses are studied. Furthermore, the characterization theorem of Moore-Penrose inverses in (Fiedler and Markham, A characterization of the Moore-Penrose inverse, Lin. Alg. Appl. 179 (1993), pp. 129-133) is extended to any reflexive g-inverse.