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...

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

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...

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...

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

{\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...

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...

} 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...

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...

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

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:...

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...

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...

the full set. (This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.) Throughout this article, pq ⁡ ( u , t 2 ) = pq ⁡ (...

_{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 )...

_{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 = ( −...

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...

else return 1; } There is another way the Jacobi and Legendre symbols differ. If the Euler's criterion formula is used modulo a composite number, the result...

follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:...

Benjamin/Cummings Publishing. ISBN 978-0-8053-7501-5. Jacobi, C. G. J. (1884), Vorlesungen über Dynamik, C. G. J. Jacobi's Gesammelte Werke (in German), Berlin: G....

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...