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P. 93
A.1. Basic results for linear equations 83
Theorem A.1.8. (a) If A is stable and R < 0 (or R ≤ 0), then the unique solution X˜ of the CALE is negative definite (or negative semi-definite, respectively).
(b) If R < 0 and if the CALE has a negative definite solution X, then A is stable.
Proof. (a) is a consequence of Theorem A.1.7.
(b) Let A∗z = λz with z ̸= 0. Multiplying (A.10) on the left and right by z∗ and z, respectively, we get
(λ¯ + λ)z∗Xz = −z∗Rz.
Since z∗Xz < 0 < −z∗Rz, this yields λ+λ = 2Rλ < 0. Hence A is stable.
In stability theory and its applications many results for CALE and DALE have been derived, see [LaTi85], Chapter 13, [LaRo95], Chapter 5 and the sur- vey article [BBC88], where additional details and proofs can be found. Further results on algebraic Sylvester and Lyapunov equations and also on their inter- relation with interpolation problems for rational matrix functions can be found in [LeRo93], [ShYa97] and [KMP00].
