Spectral abscissa explained

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

\alpha(A) . As a transformation \alpha: \Mu ^{n} → \Reals , the spectral abscissa maps a square matrix onto its largest real eigenvalue.^[2]

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:

\alpha(A) = max _{i\{} \operatorname{Re}( λ _{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 .^[3] Analogously, in control theory, the solution to the differential equation • x = Ax is stable under the same condition \alpha(A)< 0 .

See also

• Spectral radius

Notes and References

1. Deutsch . Emeric . 1975 . The Spectral Abscissa of Partitioned Matrices . Journal of Mathematical Analysis and Applications . 50 . 66–73 . CORE.
2. Burke . J. V. . Lewis . A. S. . Overton . M. L. . OPTIMIZING MATRIX STABILITY . Proceedings of the American Mathematical Society . 129 . 3 . 1635–1642.
3. Burke . James V. . Overton . Micheal L. . 1994 . DIFFERENTIAL PROPERTIES OF THE SPECTRAL ABSCISSA AND THE SPECTRAL RADIUS FOR ANALYTIC MATRIX-VALUED MAPPINGS . Nonlinear Analysis, Theory, Methods & Applications . 23 . 4 . 467–488 . Pergamon.