Spectral abscissa

Let λ₁, ..., λ_s be the (real or complex) eigenvalues of a matrix A ∈ C^{n × n}. Then its spectral abscissa is defined as:

${\displaystyle \alpha (A)=\max _{i}\{\operatorname {Re} (\lambda _{i})\}\,}$

In stability theory, a continuous system represented by matrix ${\displaystyle A}$ is said to be stable if all real parts of its eigenvalues are negative, i.e. ${\displaystyle \alpha (A)<0}$ .^[3] Analogously, in control theory, the solution to the differential equation ${\displaystyle {\dot {x}}=Ax}$ is stable under the same condition ${\displaystyle \alpha (A)<0}$ .^[2]

• Spectral radius