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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.
This polynomial is called the characteristic polynomial of A. Equation is called the characteristic equation or the secular equation of A. The fundamental theorem of algebra implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
Urbain Le Verrier (1811–1877) The discoverer of Neptune.. In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial = of a square matrix, A, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an N th-order polynomial equation in the unknown λ. This equation will have N λ distinct solutions, where 1 ≤ N λ ≤ N. The set of solutions, that is, the eigenvalues, is called the spectrum of A. [1] [2] [3]
The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation + =. In general, the solution will be n complex numbers where n is the order of the characteristic polynomial. The preceding is valid for single-input-single-output systems (SISO).
Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry , the signed n -dimensional volume of a n -dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the n -dimensional volume are ...
Given an analytic function = = and the characteristic polynomial p(x) of degree n of an n × n matrix A, the function can be expressed using long division as = () + (), where q(x) is some quotient polynomial and r(x) is a remainder polynomial such that 0 ≤ deg r(x) < n.