Discrete Lyapunov equation: simultaneous eigenvalue lower bounds

Some lower bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Lyapunov matrix equation are presented. These bounds are stronger than the majority of the relevant bounds shown in the literature. They complete some known bounds such as the extremal eigenvalues, the determinant and the trace of the solution of the above equation.

[1]  On Bounds for the Solution to the Discrete Lyapunov Matrix Equation , 1979 .

[2]  T. Mori,et al.  On the discrete Lyapunov matrix equation , 1982 .

[3]  C. Storey,et al.  Matrix methods in stability theory , 1972, The Mathematical Gazette.

[4]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[5]  J. Garloff.  Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equations and the continuous Lyapunov equation , 1986 .

[6]  N. Komaroff.  Simultaneous eigenvalue lower bounds for the Lyapunov matrix equation , 1988 .

[7]  K. Hirai,et al.  Upper and lower bounds on the solution of the algebraic Riccati equation , 1979 .

[8]  M. C. Pease,et al.  Methods of Matrix Algebra , 1965 .

[9]  T. Mori,et al.  A brief summary of the bounds on the solution of the algebraic matrix equations in control theory , 1984 .

[10]  M. Toda,et al.  On norm bounds for algebraic Riccati and Lyapunov equations , 1978 .

[11]  Myung-Joong Youn,et al.  On bounds of the Riccati and Lyapunov matrix equations , 1985 .

[12]  M. T. Tran,et al.  A note on the discrete Lyapunov and Riccati matrix equations , 1984 .

[13]  H. Weyl.  Inequalities between the Two Kinds of Eigenvalues of a Linear Transformation. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[14]  A. Horn,et al.  On the singular values of a product of completely continuous operators. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[15]  A. R. Amir-Moéz.  Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations , 1956 .