The matrices studied here are positive stable (or briefly stable). These are matrices, real or complex, whose eigenvalues have positive real parts. A theorem of Lyapunov states that A is stable if and only if there exists H ˃ 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation LA :H → AH + HA* are discussed.
1. Let C1 (A) = {AH + HA* :H ≥ 0} and C2 (A) = {H: AH+HA* ≥ 0}. The problems of determining the cones C1(A) and C2(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C1(A) is the polar of C2(A*), and it is also shown that C1 (A) = C1(A-1). The inertia assumed by matrices in C1(A) is characterized.
2. The index of dissipation of A was defined to be the maximum number of equal eigenvalues of H, where H runs through all matrices in theinterior of C2(A). Upper and lower bounds, as well as some properties of this index, are given.
3. We consider the minimal eigenvalue of the Lyapunov transform AH+HA*, where H varies over the set of all positive semi-definite matrices whose largest eigenvalue is less than or equal to one. Denote it by ψ(A). It is proved that if A is Hermitian and has eigenvalues μ1 ≥ μ2…≥ μn ˃ 0, then ψ(A) = -(μ1-μn)2/(4(μ1 + μn)). The value of ψ(A) is also determined in case A is a normal, stable matrix. Then ψ(A) can be expressed in terms of at most three of the eigenvalues of A. If A is an arbitrary stable matrix, then upper and lower bounds for ψ(A) are obtained.
