Determinants

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

theorem of solving a system of linear equations

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

square matrix with non-zero determinant

mnemonic rule for evaluating the determinant of 3x3 matrices

the matrix of all first-order partial derivatives of a vector-valued function

subclass of determinant

Mathematical concept

In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order . It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.

n×n determinant as sum of n minors weighted by cofactor from row and column not in minor

In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.

expression that describes the wave function of a multi-fermionic system

square matrix whose entries are unit fractions of special form

matrix of inner products of a set of vectors

theorem about Fibonacci numbers

matrix in which each row is rotated one position to the right from the previous row

matrix with 1/(x_i-y_j) entries

thumb | right | alt=Johann Friedrich Pfaff | Johann Friedrich Pfaff In mathematics, the determinant of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero, and when m is even, it is a nonzero polynomial of degree m/2, and is unique up to multiplication by ±1. The convention on skew-symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial. The value of this polyn

mathematics formula

theorem

top-dimensional differential form that can be defined on orientable manifolds

formula for the derivative of the determinant of a matrix

symmetric with respect to the main skew diagonal

formula for the "volume" of an n-simplex

square matrix such that the determinant of any square submatrix, including the minors, is positive

matrix formed by applying a finite list of functions pointwise to a fixed column of inputs

an analogue of the formula det(AB) = det(A) det(B), for matrices with noncommuting entrie

For two suitable matrices, A and B, I+AB and I+BA have the same determinate

Algorithm for calculating determinants

method of computing determinants

complex-valued function