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. It is sometimes denoted. As a transformation, the spectral abscissa maps a square matrix onto its largest real eigenvalue.

Matrices

Let λ₁,..., λ_s be the eigenvalues of a matrix A ∈ C^{n × n}. Then its spectral abscissa is defined as:
In stability theory, a continuous system represented by matrix is said to be stable if all real parts of its eigenvalues are negative, i.e.. Analogously, in control theory, the solution to the differential equation is stable under the same condition.