Spectral abscissa - Misplaced Pages

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). It is sometimes denoted $\alpha (A)$ . As a transformation $\alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R}$ , the spectral abscissa maps a square matrix onto its largest real eigenvalue.

Matrices

Let λ₁, ..., λ_s be the (real or complex) eigenvalues of a matrix A ∈ C. Then its spectral abscissa is defined as:

\alpha (A)=\max _{i}\{\operatorname {Re} (\lambda _{i})\}\,

In stability theory, a continuous system represented by matrix $A$ is said to be stable if all real parts of its eigenvalues are negative, i.e. $\alpha (A)<0$ . Analogously, in control theory, the solution to the differential equation ${\dot {x}}=Ax$ is stable under the same condition $\alpha (A)<0$ .

References

Deutsch, Emeric (1975). "The Spectral Abscissa of Partitioned Matrices" (PDF). Journal of Mathematical Analysis and Applications. 50: 66–73. doi:10.1016/0022-247X(75)90038-4 – via CORE.
^ Burke, J. V.; Lewis, A. S.; Overton, M. L. (2000). "Optimizing matrix stability" (PDF). Proceedings of the American Mathematical Society. 129 (3): 1635–1642. doi:10.1090/S0002-9939-00-05985-2.
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. doi:10.1016/0362-546X(94)90090-6 – via Pergamon.

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