Controllability, inertia, and stability for tridiagonal matrices
If A is an \(n\times n\) complex tridiagonal matrix and \(In A=(\pi(A),\nu(A),\delta(A))\) is the inertia of A (\(\pi\),\(\nu\),\(\delta\) being the number of eigenvalues with positive, negative and zero real parts, respectively), A is called positive stable (or just stable) if \(\pi(A)=n\) and D-stable if DA is stable for every positive diagonal matrix D. On the other hand, if B is an \(n\times n\) complex matrix, the pair (A,B) is controllable if \([B,AB,A^ 2B,...,A^{n-1}B]\) has rank n. In a previous paper the author, \textit{B. Datta} and \textit{C. Johnson} [A semi-definite Lyapunov theorem and the characterization of tridiagonal D- stable matrices. SIAM J. Algebra Discrete Math., (to appear)] characterized the real tridiagonal matrices which are D-stable. Extending the same techniques and using also results of other mathematicians about inertia and stability in relationship with controllability, the author gives criteria for the controllability of certain pairs of tridiagonal matrices, obtains some inertia results, characterizes certain matrices which are positive stable or D-stable.
scientific article; zbMATH DE number 3842060