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The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary ...
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 characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite. A characteristic function is uniformly continuous on the entire space. It is non-vanishing in a region around zero: φ(0) = 1. It is bounded: | φ(t) | ≤ 1.
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.
Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry , the signed n -dimensional volume of a n -dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the n -dimensional volume are ...
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation.
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.