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Using the language of exterior algebra, the characteristic polynomial of an matrix may be expressed as = = where is the trace of the th exterior power of , which has dimension (). This trace may be computed as the sum of all principal minors of A {\displaystyle A} of size k . {\displaystyle k.}
The trace of a linear map f : V → V can then be defined as the trace, in the above sense, of the element of V ⊗ V* corresponding to f under the above mentioned canonical isomorphism. Using an explicit basis for V and the corresponding dual basis for V * , one can show that this gives the same definition of the trace as given above.
The coefficients c i are given by the elementary symmetric polynomials of the eigenvalues of A.Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues: = = = (), where tr(A k) is the trace of the matrix A k.
The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial.This also implies that all square matrices satisfy + + () = where the coefficients are given by the elementary symmetric polynomials of the eigenvalues of A.
Characteristic polynomial, and attributes that can be derived from it: Determinant; Trace; Eigenvalues, and their algebraic multiplicities; Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used). Minimal polynomial; Frobenius normal form
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.
The trace, Tr L/K (α), is defined as the trace (in the linear algebra sense) of this linear transformation. [ 1 ] For α in L , let σ 1 ( α ), ..., σ n ( α ) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K ).
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: = [], = [].