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).^[1] It is sometimes denoted ${\displaystyle \alpha (A)}${\displaystyle \alpha (A)}. As a transformation ${\displaystyle \alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R} }${\displaystyle \alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R} }, the spectral abscissa maps a square matrix onto its largest real eigenvalue.^[2]

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

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

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

• Spectral radius

• Spectral theory
• Matrix theory
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