Spectral abscissa
Appearance
In mathematics, the spectral abscissa of a matrix or a bounded linear operator is the greatest real part of the matrix's spectrum (its set of eigenvalues).[1] It is sometimes denoted . As a transformation , the spectral abscissa maps a square matrix onto its largest real eigenvalue.[2]
Matrices
Let λ1, ..., λs be the (real or complex) eigenvalues of a matrix A ∈ Cn × n. Then its spectral abscissa is defined as:
In stability theory, a continuous system represented by matrix is said to be stable if all real parts of its eigenvalues are negative, i.e. .[3] Analogously, in control theory, the solution to the differential equation is stable under the same condition .[2]
See also
References
- ^ Deutsch, Emeric (1975). "The Spectral Abscissa of Partitioned Matrices" (PDF). Journal of Mathematical Analysis and Applications. 50: 66–73 – via CORE.
- ^ a b Burke, J. V.; Lewis, A. S.; Overton, M. L. "OPTIMIZING MATRIX STABILITY" (PDF). Proceedings of the American Mathematical Society. 129 (3): 1635–1642.
- ^ Burke, James V.; Overton, Micheal L. (1994). "DIFFERENTIAL PROPERTIES OF THE SPECTRAL ABSCISSA AND THE SPECTRAL RADIUS FOR ANALYTIC MATRIX-VALUED MAPPINGS" (PDF). Nonlinear Analysis, Theory, Methods & Applications. 23 (4): 467–488 – via Pergamon.