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).^[1] 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.^[2]

Matrices

Let λ₁, ..., λ_s be the (real or complex) eigenvalues of a matrix A ∈ C^{n × 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. $\alpha (A)<0$ .^[3] Analogously, in control theory, the solution to the differential equation ${\dot {x}}=Ax$ is stable under the same condition $\alpha (A)<0$ .^[2]

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.
^ ^a ^b 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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[1] 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.

[:0-2] 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.

[3] 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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v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

Matrices

See also

References