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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,
The objective is to calculate the coefficients c k of the characteristic polynomial of the n×n matrix A, () = = ,where, evidently, c n = 1 and c 0 = (−1) n det A. The coefficients c n-i are determined by induction on i, using an auxiliary sequence of matrices
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
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 equation is linear and homogeneous, and has constant ...
Computing the characteristic polynomial and choosing a suitable feedback matrix can be a challenging task, especially in larger systems. One way to make computations easier is through Ackermann's formula. For simplicity's sake, consider a single input vector with no reference parameter r, such as
A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial. The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions e pt of the system that are stable ( bounded ).
Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powers and their traces. This computation requires computing the traces of matrix powers A k {\displaystyle \mathbf {A} ^{k}} and solving a triangular system of equations.