𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Common solution to the Lyapunov equation for 2 × 2 complex matrices

✍ Scribed by Thomas J. Laffey; Helena Šmigoc

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
182 KB
Volume
420
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

✦ Synopsis

In this work we solve the problem of a common solution to the Lyapunov equation for 2 × 2 complex matrices. We show that necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for 2 × 2 complex matrices A and B is that matrices (A + iαI )(B + iβI ) and (A + iαI ) -1 (B + iβI ) have no negative real eigenvalues for all α, β ∈ R. We show how these results relate to a special class of 4 × 4 real matrices.

📜 SIMILAR VOLUMES

B-Orbits in Solutions to the Equation X2
B-Orbits in Solutions to the Equation X2 = 0 in Triangular Matrices
✍ Anna Melnikov 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 83 KB

Consider I n K , the set of solutions to the equation X 2 = 0 in n × n strictly uppertriangular matrices over field K. In this paper we construct the bijection from the set of adjoint orbits in I n K onto the set of involutions in symmetric group S n . For a finite field K we compute the number of p

A note on the representations for the Dr
A note on the representations for the Drazin inverse of 2 × 2 block matrices
✍ Xiezhang Li; Yimin Wei 📂 Article 📅 2007 🏛 Elsevier Science 🌐 English ⚖ 126 KB

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix M = A B C D in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of

The intersection of left (right) spectra
The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices
✍ Yuan Li; Xiu-Hong Sun; Hong-Ke Du 📂 Article 📅 2006 🏛 Elsevier Science 🌐 English ⚖ 141 KB

When A ∈ B(H) and B ∈ B(K) are given, we denote by M C the operator matrix acting on the infinite- . In this paper, for given A and B, the sets C∈B l (K,H) σ l (M C ), C∈Inv(K,H) σ l (M C ) and C∈Inv(K,H) σ l (M C ) are determined, where σ l (T ), B l (K, H) and Inv(K, H) denote, respectively, the