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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.
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.
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.
Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary axis (i.e. the line z = ic where i is the imaginary unit and c is a real number).Let us define real polynomials P 0 (y) and P 1 (y) by f(iy) = P 0 (y) + iP 1 (y), respectively the real and imaginary parts of f on the imaginary line.
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: = [], = [].
[3] [4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. [1] Such a differential equation, with y as the dependent variable , superscript ( n ) denoting n th - derivative , and a n , a n − 1 , ..., a 1 , a 0 as constants,
which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator (() /), such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic ...
By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an ƒ with no roots on the imaginary line, p + q = n). Thus, we have the condition that ƒ is a (Hurwitz) stable polynomial if and only if p − q = n (the proof is given below).