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 $\alpha :{\mathrm{M}}^n\to \mathbb{ℝ}$, 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:

$\alpha (A)=\underset{i}{\max }\{\mathrm{Re}({\lambda }_i)\}$

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. $\alpha (A)<0$. Analogously, in control theory, the solution to the differential equation $\dot{x}=Ax$ is stable under the same condition $\alpha (A)<0$.

See also

• Spectral radius