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Spectral abscissa

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).{{Cite journal |last=Deutsch |first=Emeric |date=1975 |title=The Spectral Abscissa of Partitioned Matrices |url=https://core.ac.uk/download/pdf/82047336.pdf |journal=Journal of Mathematical Analysis and Applications |volume=50 |pages=66–73 |doi=10.1016/0022-247X(75)90038-4 |via=CORE}} It is sometimes denoted \alpha(A). As a transformation \alpha: \Mu^n \rightarrow \Reals , the spectral abscissa maps a square matrix onto its largest real eigenvalue.{{Cite journal |last1=Burke |first1=J. V. |last2=Lewis |first2=A. S. |last3=Overton |first3=M. L. |title=Optimizing matrix stability |url=https://www.ams.org/journals/proc/2001-129-06/S0002-9939-00-05985-2/S0002-9939-00-05985-2.pdf |journal=Proceedings of the American Mathematical Society |date=2000 |volume=129 |issue=3 |pages=1635–1642|doi=10.1090/S0002-9939-00-05985-2 }}

Matrices

Let λ1, ..., λs be the (real or complex) eigenvalues of a matrix A ∈ Cn × n. 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.{{Cite journal |last1=Burke |first1=James V. |last2=Overton |first2=Micheal L. |date=1994 |title=Differential properties of the spectral abscissa and the spectral radius for analytic matrix-valued mappings |url=https://sites.math.washington.edu/~burke/papers/reprints/22-diff-spec-abs-rad1994.pdf |journal=Nonlinear Analysis, Theory, Methods & Applications |volume=23 |issue=4 |pages=467–488 |doi=10.1016/0362-546X(94)90090-6 |via=Pergamon}} Analogously, in control theory, the solution to the differential equation \dot{x}=Ax is stable under the same condition \alpha(A)<0.

See also

References

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{{Functional analysis}}

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Category:Spectral theory

Category:Matrix theory

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