DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Hurwitz matrices and the continuous Lyapunov equation

Definition

Let $F \in {R, C}$ , $n \geq 1$ , and $A \in F^{n \times n}$ . The matrix $A$ is Hurwitz if, when its eigenvalues are considered over $C$ , every eigenvalue has strictly negative real part. For $Q, X \in F^{n \times n}$ with $Q = Q^{*}$ , the continuous Lyapunov equation with coefficient $A$ and forcing $Q$ is

A X + X A^{*} + Q = 0.

Here $^{*}$ is transpose for real matrices and conjugate transpose for complex matrices. The notations $Q ⪰ 0$ and $Q ≻ 0$ mean that $Q$ is non-negative or positive as a self-adjoint operator in the sense of Non-negative and positive operators: respectively $v^{*} Q v \geq 0$ for all $v$ , or $v^{*} Q v > 0$ for every nonzero $v$ .

Depends on

Non-negative and positive operators

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Valeria Simoncini, Computational Methods for Linear Matrix Equations, SIAM Review 58(3), 2016 (standard reference, not scraped)
Elias Jarlebring, Numerical Methods for Lyapunov Equations, lecture notes in numerical linear algebra, 2017 (standard reference, not scraped)