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 ${\displaystyle \alpha (A)}$. As a transformation ${\displaystyle \alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R} }$, 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:

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

In stability theory, a continuous system represented by matrix ${\displaystyle A}$ is said to be stable if all real parts of its eigenvalues are negative, i.e. ${\displaystyle \alpha (A)<0}$.^[3] Analogously, in control theory, the solution to the differential equation ${\displaystyle {\dot {x}}=Ax}$ is stable under the same condition ${\displaystyle \alpha (A)<0}$.^[2]

See also

• Spectral radius

References

1. Deutsch, Emeric (1975). "The Spectral Abscissa of Partitioned Matrices" (PDF). Journal of Mathematical Analysis and Applications. 50: 66–73 – via CORE.
2. Burke, J. V.; Lewis, A. S.; Overton, M. L. "OPTIMIZING MATRIX STABILITY" (PDF). Proceedings of the American Mathematical Society. 129 (3): 1635–1642.
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 – via Pergamon.