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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,
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 ...
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 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 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: = [], = [].
For example, if A is a multiple aI n of the identity matrix, then its minimal polynomial is X − a since the kernel of aI n − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a) n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space).
That is, the characteristic polynomial must factor completely into linear factors; must be an algebraically closed field. For example, if A {\displaystyle A} has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values .