Page 91 - Textos de Matemática Vol. 36

A.1. Basic results for linear equations
81
Theorem A.1.3. (i) The algebraic Sylvester equation XN + MX + R = 0
has a unique solution X if and only if
σ(N) ∩ σ(−M) = ∅.
(ii) The algebraic Stein equation
S − NSA = R has a unique solution S ∈ Cn×n if and only if
λjμk ̸= 1 for all λj ∈ σ(A), μk ∈ σ(N).
Proof. Writing ImXN + MXIn instead of XN + MX we infer from Remark
A.1.2, (iv), that (A.11) is equivalent to
KX⃗ :=(NT ⊗Im +In ⊗M)X⃗ =−R⃗,
where the coefficient matrix K has, according to Remark A.1.2, (iii), the eigen- values
λj +μk with λj ∈σ(N)=σ(NT), μk ∈σ(M).
Hence K is invertible (and (A.11) has a unique solution) if and only if (A.12)
holds. (ii) is proved analogously. As an immediate consequence of this theorem we get
Corollary A.1.4. (i) The algebraic Lyapunov equation (A.10) has a unique solution X if and only if σ(A)∩σ(−A∗) = ∅; in particular this means that A must not have purely imaginary eigenvalues.
(ii) Assume that R = R∗. The discrete-time algebraic Lyapunov equation (or Hermitian Stein equation)
X = A∗XA + R (DALE) has a unique solution X if and only if A has no eigenvalues on the unit
circle. (Notice that this still remains valid for R ̸= R∗.)
Theorem A.1.5. The (unique) solution of the Sylvester differential equation
is defined by
X˙ = M(t)X + XN(t) + R(t), X(t0) = X0 (A.15)
t t0
X(t) = ΦM(t,t0)X0Φ∗N∗(t,t0)+
ΦM(t,τ)R(τ)Φ∗N∗(t,τ)dτ, t ∈ R. (A.16)
(A.11)
(A.12)
(A.13)
(A.14)