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). 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.

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:

:$\alpha(A) = \max_i\ \,$

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 $\dot=Ax$ is stable under the same condition $\alpha(A)<0$.

See also

* Spectral radius

References

Spectral theory
Matrix theory

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