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Solving the characteristic equation for its roots, r 1, ..., r n, allows one to find the general solution of the differential equation. The roots may be real or complex, as well as distinct or repeated. If a characteristic equation has parts with distinct real roots, h repeated roots, or k complex roots corresponding to general solutions of y D ...
which is the characteristic equation of the recurrence relation. Solve for to obtain the two roots , : these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution
A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied: P(s) is real when s is real. The roots of P(s) have real parts which are zero or negative. Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems.
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. A linear transformation's ...
The recurrence is stable, meaning that the iterates converge asymptotically to a fixed value, if and only if the eigenvalues (i.e., the roots of the characteristic equation), whether real or complex, are all less than unity in absolute value.
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b being real numbers, then its complex conjugate a − bi is also a root of P. [1]
Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable.
For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial).