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 . As a transformation , the spectral abscissa maps a square matrix onto its largest real eigenvalue.[2]
Let λ1, ..., λs be the (real or complex) eigenvalues of a matrix A ∈ Cn × 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. .[3] Analogously, in control theory, the solution to the differential equation is stable under the same condition .[2]