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Here, is the indeterminate of the polynomial and is the identity matrix of the same size as . By means of this polynomial, determinants can be used to find the eigenvalues of the matrix A {\displaystyle A} : they are precisely the roots of this polynomial, i.e., those complex numbers λ {\displaystyle \lambda } such that
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi-with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. [6]
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.
An explanation of the sigma (Σ) summation notation. ... For getting an algorithm that can be implemented and can solve systems of polynomial equations and ...
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
In such cases (and only in such cases), it is possible to obtain the pseudoinverse as a polynomial in . A polynomial p ( t ) {\displaystyle p(t)} such that A + = p ( A ) {\displaystyle A^{+}=p(A)} can be easily obtained from the characteristic polynomial of A {\displaystyle A} or, more generally, from any annihilating polynomial ...