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 . As a transformation , the spectral abscissa maps a square matrix onto its largest real eigenvalue.
Let û<sub>1</sub>, ..., û<sub>s</sub> be the (real or complex) eigenvalues of a matrix A â C<sup>n àn</sup>. 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 .