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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 definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra = (), but also includes several further cases including the determinant of a quaternion,
Using the Leibniz formula for determinants, the left-hand side of equation is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1) n λ n. This polynomial is called the characteristic polynomial of A.
A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton ...
A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
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 ...
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: = [], = [].
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