週期矩陣常常出現在動態系統的學習上,而保秩的矩陣在微分代數系統也是很重要地.在本篇論文中我們考慮以下平滑及週期的線性矩陣方程其係數為保秩的線性矩陣係數.
(1.1) A(t)x(t)=b(t),
(1.2) A(t)X(t)B(t)=E(t),
(1.3) A(t)X(t) + Y(t)B(t)=C(t),
(1.4) A(t)X(t)B(t) + C(t)Y(t)D(t)=E(t).
因為它們可能無解所以我們有興趣的是以下平滑及週期的最小l_2-解的問題.
(1.1a) min||A(t)x(t)-b(t)||_2
(1.2a) min||A(t)X(t)B(t)-E(t)||_2
(1.3a) min||A(t)X(t)+Y(t)B(t)-C(t)||_2
(1.4a) min||A(t)X(t)B(t)+C(t)Y(t)D(t)-E(t)||_2

Periodic matrices arise quite often in the study of dynamics.
The matrices with constant rank is important in applications related to differential algebraic system.In this paper we consider the following smooth and periodic linear matrix equations with constant rank matrix coefficients respectively.
(1.1) A(t)x(t)=b(t),
(1.2) A(t)X(t)B(t)=E(t),
(1.3) A(t)X(t) + Y(t)B(t)=C(t),
(1.4) A(t)X(t)B(t) + C(t)Y(t)D(t)=E(t).
Because they may be inconsistent (i.e., have no solution),
we are interesting in the following smooth and periodic minimal l_2-solution problems respectively.
(1.1a) min||A(t)x(t)-b(t)||_2
(1.2a) min||A(t)X(t)B(t)-E(t)||_2
(1.3a) min||A(t)X(t)+Y(t)B(t)-C(t)||_2
(1.4a) min||A(t)X(t)B(t)+C(t)Y(t)D(t)-E(t)||_2

[1] J. K. Baksalary and R. Kala, The matrix equation AX-YB=C, Linear Algebra Appl., 25:41-43, 1979.
[2] J. K. Baksalary and R. Kala, The matrix equation
AXB+CYD=E, Linear Algebra Appl., 30:141-147, 1980.
[3] K. E. Brebnan, S. L. Campbell and L. R. Petzold, Numerical solution of IVPs in DAEs., North-Holland, New York, 1989.
[4] S. Campbell, Numerical solution of higher index linear time varying singular systems of DAEs., SIAM J. Scient. Stat. Comp. 6, 334-348, 1988.
[5] J.-L. Chern, The smooth SSVD of periodic complex symmetric metrices, Preprint.
[6] J.-L. Chern and L. Dieci, Smoothness and periodicity of some matrix decompositions, to appear in SIAM J. Matrix Anal. Appl..
[7] K.-W. E. Chu, Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra and its Appl., 88/89:83-98, 1987.
[8] L. Dieci and T. Eirola, On smooth decompositions of matrices, SIAM J. Matrix Anal. Appl., 20:800-819, 1999.
[9] F. R. Gantmacher, The theory of Matrices Vol II, Chelsea, New York, 1959.
[10] G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, 2nd edition, 1989.
[11] J. K. Hale, Ordinary Differential Equations, Krieger Publishing Co, Malabar, 1980.
[12] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
[13] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
[14] P. Kunkel and V. Mehrmann, Canonical forms for linear DAEs with variable coefficients, J. Comp. Appl. Math. 56:225-251, 1994.
[15] E. V. Mamontov, Some properties of a system of first order ordinary differential nonlinear equations with a singular
matrix of constant rank in front of the vector of the derivatives, Differentsialnye Uravneniya, 24:1055-1058, 1988.
[16] M. Z. Nashed (Ed), Generalized Inverses and Applications, New York: Academic, 1976.
[17] C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer Anal., 18:398-405, 1981.
[18] Y. Sibuya, Some global properties of matrices of functions of one variable, Math. Annal., 161:67-77, 1965.
[19] G. W. Stewart, Introduction to Matrix Computations, New York: Academic Press, 1973.
[20] K. Zietak, On a particular case of the inconsistent linear matrix equation AX+YB=C, Linear Algebra and its Appl., 66:249-258, 1985.