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In mathematics, the characteristic equation (or auxiliary equation [1]) is an algebraic equation of degree n upon which depends the solution of a given n th-order differential equation [2] or difference equation. [3] [4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and ...
Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data ...
The characteristic of a ring R is the natural number n such that n is the kernel of the unique ring homomorphism from to R. [a]The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring /, which is the image of the above homomorphism.
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, a technique for solving partial differential equations
Polynomial equation. Linear equation; Quadratic equation; Cubic equation; Biquadratic equation; Quartic equation; Quintic equation; Sextic equation; Characteristic ...
The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those ...