A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provi...A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .展开更多
We show that the zeros of a trigonometric polynomial of degree N with the usual(2N+1)terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements M_(jk).This matrix M is a...We show that the zeros of a trigonometric polynomial of degree N with the usual(2N+1)terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements M_(jk).This matrix M is a multiplication matrix in the sense that,after first defining a vector φwhose elements are the first 2N basis functions,Mφ=2cos(t)φ.This relationship is the eigenproblem;the zeros tk are the arccosine function of λ_(k)/2 where theλk are the eigenvalues of M.We dub this the“Fourier Division Companion Matrix”,or FDCM for short,because it is derived using trigonometric polynomial division.We show through examples that the algorithm computes both real and complex-valued roots,even double roots,to near machine precision accuracy.展开更多
We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial <i>P</i>(<i>X</i>) over the field C[<i>X</i>]. From a <i>n</i>...We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial <i>P</i>(<i>X</i>) over the field C[<i>X</i>]. From a <i>n</i>×<i>n</i> Fiedler companion matrix <i>C</i>, sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix <i>L<sub>r</sub></i>, supposed to provide a good estimation of the roots. By application of Gershgorin’s theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds. According to the starting formel of <i>L<sub>r</sub></i>, we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.展开更多
In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are est...In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.展开更多
基金supported by the Major Program of National Nat-ural Science Foundation of China (No. 60710002) Program for Changjiang Scholars and Innovative Research Team in University
文摘A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .
基金supported by the National Science Foundation through grant OCE 1059703.
文摘We show that the zeros of a trigonometric polynomial of degree N with the usual(2N+1)terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements M_(jk).This matrix M is a multiplication matrix in the sense that,after first defining a vector φwhose elements are the first 2N basis functions,Mφ=2cos(t)φ.This relationship is the eigenproblem;the zeros tk are the arccosine function of λ_(k)/2 where theλk are the eigenvalues of M.We dub this the“Fourier Division Companion Matrix”,or FDCM for short,because it is derived using trigonometric polynomial division.We show through examples that the algorithm computes both real and complex-valued roots,even double roots,to near machine precision accuracy.
文摘We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial <i>P</i>(<i>X</i>) over the field C[<i>X</i>]. From a <i>n</i>×<i>n</i> Fiedler companion matrix <i>C</i>, sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix <i>L<sub>r</sub></i>, supposed to provide a good estimation of the roots. By application of Gershgorin’s theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds. According to the starting formel of <i>L<sub>r</sub></i>, we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.
基金This work was supported by the Chinese Outstanding Youth Foundation(No.69925308)Program for Changjiang Scholars and Innovative ResearchTeam in University.
文摘In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.
