On the Controllability of Matrix Pairs $(A,K)$ with K Positive Semidefinite
Copyright policy )On the Controllability of Matrix Pairs $(A,K)$ with K Positive Semidefinite
A well-known result of Chen and Wimmer is: If A, H and K are \(n\times n\) matrices such that H and K are Hermitian, K is positive semidefinite and \(K=AH+HA^*\) and if (A,K) is controllable, then A has no eigenvalues on the imaginary axis and H is nonsingular. Moreover the numbers of eigenvalues of A with positive and negative real parts equal respectively the numbers of positive and negative eigenvalues of H. It is also known that the converse statement \([K=AH+HA^*\geq 0\) for some Hermitian H and A has no eigenvalues on the imaginary axis\(\Rightarrow (A,K)\) is controllable] is false in general. The authors show that the converse does hold if one amends the conclusion to the assertion that a related pair of matrices (A,\~K) be controllable. Also they consider more general partitionings of the eigenvalues of A.
- University of Wisconsin–Oshkosh United States
- Oregon State University United States
- Northwestern University United States
- San Diego State University United States
Controllability, Eigenvalues, singular values, and eigenvectors, Matrix equations and identities, Lyapunov matrix maps, Inequalities involving eigenvalues and eigenvectors, eigenvalue location
Controllability, Eigenvalues, singular values, and eigenvectors, Matrix equations and identities, Lyapunov matrix maps, Inequalities involving eigenvalues and eigenvectors, eigenvalue location
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