Characteristic Equation (other)

	Characteristic Equation (other) Characteristic equation may refer to: * Characteristic equation (calculus), used to solve linear differential equations * Characteristic equation, the equation obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping * Method of characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial d ..., a technique for solving partial differential equations See also * Characteristic (other) {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Characteristic Equation (calculus) In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree upon which depends the solution of a given th- order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with as the dependent variable, superscript denoting ''n''th-derivative, and as constants, :a_y^ + a_y^ + \cdots + a_y' + a_y = 0, will have a characteristic equation of the form :a_r^ + a_r^ + \cdots + a_r + a_ = 0 whose solutions are the roots from which the general solution can be formed. Analogously, a linear difference equation of the form :y_=b_1y_ + \cdots + b_ny_ has characteristic equation :r^n - b_1r^ - \cdots - b_n =0, discussed in more detail at Linear recurrence with constant coefficients#Solution to homogeneous case. The characteristic roots (roots of the characteristic equation) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Characteristic Polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Method Of Characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. Characteristics of first-order partial differential equation For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]