# 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 [math]\displaystyle{ \alpha(A) }[/math]. As a transformation [math]\displaystyle{ \alpha: \Mu^n \rightarrow \Reals }[/math], the spectral abscissa maps a square matrix onto its largest real eigenvalue.^{[2]}

## Matrices

Let λ_{1}, ..., λ_{s} be the (real or complex) eigenvalues of a matrix *A* ∈ **C**^{n × n}. Then its spectral abscissa is defined as:

- [math]\displaystyle{ \alpha(A) = \max_i\{ \operatorname{Re}(\lambda_i) \} \, }[/math]

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

## See also

## References

- ↑ Deutsch, Emeric (1975). "The Spectral Abscissa of Partitioned Matrices".
*Journal of Mathematical Analysis and Applications***50**: 66–73. https://core.ac.uk/download/pdf/82047336.pdf. - ↑
^{2.0}^{2.1}Burke, J. V.; Lewis, A. S.; Overton, M. L.. "OPTIMIZING MATRIX STABILITY".*Proceedings of the American Mathematical Society***129**(3): 1635–1642. https://www.ams.org/journals/proc/2001-129-06/S0002-9939-00-05985-2/S0002-9939-00-05985-2.pdf. - ↑ Burke, James V.; Overton, Micheal L. (1994). "DIFFERENTIAL PROPERTIES OF THE SPECTRAL ABSCISSA AND THE SPECTRAL RADIUS FOR ANALYTIC MATRIX-VALUED MAPPINGS".
*Nonlinear Analysis, Theory, Methods & Applications***23**(4): 467–488. https://sites.math.washington.edu/~burke/papers/reprints/22-diff-spec-abs-rad1994.pdf.

Original source: https://en.wikipedia.org/wiki/Spectral abscissa.
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