In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If...

p(z\mathbf {I} ,\mathbf {A} )}{p(z)}}.} The adjugate also appears in Jacobi's formula for the derivative of the determinant. If A(t) is continuously differentiable...

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde...

Jacob Jacobi and published in its Latinized form as Carolus Gustavus Jacobus Jacobi. He is sometimes referred to as C. G. J. Jacobi. One of Jacobi's greatest...

diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of...

operations Jacobi's formula for the derivative of the determinant of a matrix Jacobi triple product, an identity in the theory of theta functions Jacobi's theorem...

similar. The trace is related to the derivative of the determinant (see Jacobi's formula). The trace of an n × n square matrix A is defined as: 34  tr ⁡ ( A...

Carl Gustav Jacob Jacobi". Journal für die reine und angewandte Mathematik (in German). 52: 193–217. Darwin, G. H. (1886). "On Jacobi's figure of equilibrium...

} This is but the trace of the defining equation for B by dint of Jacobi's formula, ∂ p A ( λ ) ∂ λ = p A ( λ ) ∑ m = 0 ∞ λ − ( m + 1 ) tr ⁡ A m = p A...

{\displaystyle \tau \mapsto {-1/\tau }} and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of...

field Jacobi's four-square theorem Jacobi form Jacobi's formula Jacobi group Jacobian ideal Jacobi identity Jacobi integral Jacobi's logarithm Jacobi method...

In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of...

is everywhere differentiable. Its derivative can be expressed using Jacobi's formula: d det ( A ) d α = tr ⁡ ( adj ⁡ ( A ) d A d α ) . {\displaystyle {\frac...

constant, but the Magnus series gives the solution as an infinite sum. By Jacobi's formula, for any complex square matrix the following trace identity holds:...

function: det D 2 u = f . {\displaystyle \det D^{2}u=f.} As follows from Jacobi's formula for the derivative of a determinant, this equation is elliptic if f...

term and simply obtain at events not in the closure of the boundary. Jacobi's formula, the rule for differentiating a determinant, gives: δ g = δ det ( g...

Gauss and C.G.J. Jacobi. The triangle form of the area formula can be considered to be a special case of Green's theorem. The area formula can also be applied...

V\rightarrow U\subset {\mathfrak {g}}.} An important corollary of Jacobi's formula then is log ⁡ ( det ( A ) ) = t r ( log ⁡ A )   . {\displaystyle...

Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ,dn) with r = x 2 + y 2 {\textstyle r={\sqrt {x^{2}+y^{2}}}} Equivalently, Jacobi's elliptic...

_{r}(A^{-1})=(\det A)^{-1}C_{r}(A).} A concrete consequence of this is Jacobi's formula for the minors of an inverse matrix: det ( A − 1 ) J c , I c = ( −...

_{l=1}^{n}lk_{l}=n-i.} See, e.g., p. 54 of Brown 1994, which solves Jacobi's formula, ∂ p ( λ ) ∂ λ = p ( λ ) ∑ m = 0 ∞ λ − ( m + 1 ) tr ⁡ A m = p ( λ )...

In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly...

system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Jacobi's proof relies on Euler's pentagonal number theorem...

x_{n})}}.} This is known as the bialternant formula of Jacobi. It is a special case of the Weyl character formula. This is a symmetric function because the...

In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their...

systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The first actual explicit formula, with all numerical coefficients, is due...

. The asymptotics of the Jacobi polynomials near the points ± 1 {\displaystyle \pm 1} is given by the Mehler–Heine formula lim n → ∞ n − α P n ( α ,...

introduced the symbol. There is another way the Jacobi and Legendre symbols differ. If the Euler's criterion formula is used modulo a composite number, the result...

Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio...

In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first...