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The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis).
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 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]
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
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).
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
The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by () = ().The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix itself, the result is the zero matrix: () =.