Characteristic Equation

characteristic equation [‚kar·ik·tə′ris·tik i′kwā·zhən]

(mathematics)

Any equation which has a solution, subject to specified boundary conditions, only when a parameter occurring in it has certain values.

Specifically, the equationAu= λu, which can have a solution only when the parameter λ has certain values, whereAcan be a square matrix which multiplies the vectoru, or a linear differential or integral operator which operates on the functionu, or in general, any linear operator operating on the vectoruin a finite or infinite dimensional vector space. Also known as eigenvalue equation.

An equation which sets the characteristic polynomial of a given linear transformation on a finite dimensional vector space, or of its matrix representation, equal to zero.

(physics)

An equation relating a set of variables, such as pressure, volume, and temperature, whose values determine a substance's physical condition.

(plasma physics)

An equation whose solutions give the frequencies and modes of those perturbations of a hydromagnetic system which decay or grow exponentially in time, and indicate regions of stability of such a system.

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Characteristic Equation 

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} + · · · + S_n = 0. Here, S₁ = a₁₁ + a₂₂ + · · · + a_nn 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 S_{n – 1} are defined correspondingly; and S_n is the determinant of the matrix A.

The roots λ₁, λ₂, . . ., λ_n of the characteristic equation are called the eigenvalues of A. If A is real symmetric (more generally, Hermitian symmetric), then the λ_k are real. If A is real and skew symmetric, then the λ_k are pure imaginary. If A is orthogonal (more generally, unitary), then all | λ_k| = 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′ + a_ny = 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 + a₁λ^{n – 1} + · · · + a_{n – 1} λ + a_n = 0. We are led to this equation if we look for solutions of the given differential equation that have the form y = ce^λx. 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.