Search results
Results from the WOW.Com Content Network
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
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: = [], = [].
Bahasa Indonesia; Italiano; עברית ... the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix ... which is a monic ...
Graph of the polynomial function x 4 + x 3 – x 2 – 7x/4 – 1/2 (in green) together with the graph of its resolvent cubic R 4 (y) (in red). The roots of both polynomials are visible too. In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums of the roots: replace each e i by a i and each p k by s k. Expressing complete homogeneous symmetric polynomials in terms of power sums
The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials (). Its existence can be shown using the notion of holonomic D-modules . Kashiwara (1976) proved that all roots of the Bernstein–Sato polynomial are negative rational numbers .
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 ...
If x is an algebraic number then a n x is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where a n x n is the highest-degree term of p(x). The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients.