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characteristic equation[‚kar·ik·tə′ris·tik i′kwā·zhən]
in mathematics. (1) The characteristic equation of a matrix is the algebraic equation
The determinant on the left-hand side of the characteristic equation is obtained by subtracting λ from the diagonal elements of the matrix . This determinant is a polynomial in λ and is called the characteristic polynomial.
The explicit form of the equation is (–λ)n + S1(–λ)n – 1 + S2(–λ)n – 2 + · · · + Sn = 0. Here, S1 = a11 + a22 + · · · + ann is the trace of the matrix; S2, is the sum of all minors of order two, that is, of all minors
where i < k; S3 through Sn – 1 are defined correspondingly; and Sn is the determinant of the matrix A.
The roots λ1, λ2, . . ., λn of the characteristic equation are called the eigenvalues of A. If A is real symmetric (more generally, Hermitian symmetric), then the λk are real. If A is real and skew symmetric, then the λk are pure imaginary. If A is orthogonal (more generally, unitary), then all | λk| = 1.
Characteristic equations are encountered in a great many areas of mathematics, physics, mechanics, and engineering. Because of their application in astronomy to the determination of secular perturbations of planets, they are also called secular equations.
(2) The characteristic equation of a linear differential equation with constant coefficients a0y(n) + a1y(n – 1) + · · · + a(n – 1)y′ + any = 0 is the algebraic equation obtained from this differential equation by replacing y and its derivatives by suitable powers of λ, that is, the equation a0λn + a1λn – 1 + · · · + an – 1 λ + an = 0. We are led to this equation if we look for solutions of the given differential equation that have the form y = ceλx. For a system of linear differential equations
the characteristic equation is
and thus coincides with the characteristic equation of the matrix whose elements are the coefficients of the equations of the system.