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 formula_1. As a transformation formula_2, 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:
formula_3
In stability theory, a continuous system represented by matrix formula_4 is said to be stable if all real parts of its eigenvalues are negative, i.e. formula_5. Analogously, in control theory, the solution to the differential equation formula_6 is stable under the same condition formula_5.
References.
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