Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore. A familiar result states that each nilpotent operator on a finite dimensional complex Hubert space is similar to its adjoint. One proof proceeds by showing that both a nilpotent operat...

The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA∗ and A∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA and AA. More generally, we consider the matrix triple (A,G, Ĝ), where G ∈ Cm×m, Ĝ ∈ Cn×n ar...

In this paper, we propose a new adaptive observer for a class of state-a¢ ne systems. More speci…cally we consider state a¢ ne systems that are in a special observable canonical form called the Generalised Jordan Observable Canonical (GJOC) form. The observer is then employed to estimate the biomass concentration in a bioreactor, using the measurements of the substrate concentration. Simulation...

The idea for nding a basis relates to the proof of why a Jordan canonical form exists. What we seek to do is nd a largest possible set of chains (or cycles) of the form {x, (T −λkI)(x), . . . , (T −λkI)(x)} which are linearly independent. By the proof of Jordan canonical form, the number and lengths of these chains can be found from the numbers d0, . . . , d`k . Indeed, let λ = λk be a xed eige...

Under consideration of this paper is the application Jordan canonical form and symplectic matrix to two conformable fractional differential models. One new vector conduction equation which reduced by using coefficient solved exactly, other dynamical system with Hamilton matrix, derived constructing Euler-Lagrange variational principle. It shown that can be used deal some systems in mathematical...