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 (its set of eigenvalues). It is sometimes denoted [math]\displaystyle{ \alpha(A) }[/math]. As a transformation [math]\displaystyle{ \alpha: \Mu^n \rightarrow \Reals }[/math], 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:

[math]\displaystyle{ \alpha(A) = \max_i\{ \operatorname{Re}(\lambda_i) \} \, }[/math]

In stability theory, a continuous system represented by matrix [math]\displaystyle{ A }[/math] is said to be stable if all real parts of its eigenvalues are negative, i.e. [math]\displaystyle{ \alpha(A)\lt 0 }[/math]. Analogously, in control theory, the solution to the differential equation [math]\displaystyle{ \dot{x}=Ax }[/math] is stable under the same condition [math]\displaystyle{ \alpha(A)\lt 0 }[/math].

See also

• Spectral radius