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, ..., λs be the (real or complex) eigenvalues of a matrix A ∈ Cn × 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. $$\alpha(A)<0$$. Analogously, in control theory, the solution to the differential equation $$\dot{x}=Ax$$ is stable under the same condition $$\alpha(A)<0$$.