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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 A is a differentiable map from the real numbers to n × n matrices, then where tr(X) is the trace of the matrix X. (The latter equality only holds if A(t) is invertible.) As a special case, Equivalently, if dA stands for the differential of A, the general formula is The formula is named after the mathematician Carl Gustav Jacob Jacobi.

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rdf:type
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  • Fórmula de Jacobi (ca)
  • Formula di Jacobi (it)
  • Jacobi's formula (en)
  • Формула Якоби (ru)
  • 雅可比公式 (zh)
rdfs:comment
  • En càlcul matricial, la fórmula de Jacobi expressa la derivada del determinant d'una matriu (quadrada) A en funció de la seva matriu adjunta i de la seva derivadaː , on tr(X) és la traça de la matriu X. Com a cas especial, De forma equivalent, si dA representa el diferencial d'A, la fórmula general és La següent relació útil connecta la traça amb el determinant de l'exponencial de la matriu associada: La fórmula rep el seu nom del matemàtic Carl Gustav Jacob Jacobi. (ca)
  • 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 A is a differentiable map from the real numbers to n × n matrices, then where tr(X) is the trace of the matrix X. (The latter equality only holds if A(t) is invertible.) As a special case, Equivalently, if dA stands for the differential of A, the general formula is The formula is named after the mathematician Carl Gustav Jacob Jacobi. (en)
  • Формула Якоби — формула, связывающая определитель матрицы, удовлетворяющей дифференциальному уравнению, в начале интервала интегрирования с определителем матрицы в конце интервала интегрирования. (ru)
  • 在矩阵微积分中，雅可比公式（Jacobi's formula）把矩阵 的行列式的导数表达为 的伴随矩阵与 本身导数的乘积的跡。 若 是从实数到 矩阵的可微映射，则 。 其中 為矩阵 的跡。 (zh)
  • In matematica, la formula di Jacobi, che prende il nome dal matematico C. G. J. Jacobi, esprime la derivata del determinante di una matrice attraverso la matrice dei cofattori (o matrice dei complementi algebrici) di e della derivata di stessa. Il determinante di una matrice può infatti considerarsi una funzione polinomiale: quindi essa è differenziabile e il suo differenziale può essere espresso mediante la formula di Jacobi: dove denota la trasposta della matrice dei cofattori (detta anche matrice aggiunta e denotata come ), mentre è la traccia. (it)
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  • En càlcul matricial, la fórmula de Jacobi expressa la derivada del determinant d'una matriu (quadrada) A en funció de la seva matriu adjunta i de la seva derivadaː , on tr(X) és la traça de la matriu X. Com a cas especial, De forma equivalent, si dA representa el diferencial d'A, la fórmula general és La següent relació útil connecta la traça amb el determinant de l'exponencial de la matriu associada: La fórmula rep el seu nom del matemàtic Carl Gustav Jacob Jacobi. (ca)
  • 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 A is a differentiable map from the real numbers to n × n matrices, then where tr(X) is the trace of the matrix X. (The latter equality only holds if A(t) is invertible.) As a special case, Equivalently, if dA stands for the differential of A, the general formula is The formula is named after the mathematician Carl Gustav Jacob Jacobi. (en)
  • In matematica, la formula di Jacobi, che prende il nome dal matematico C. G. J. Jacobi, esprime la derivata del determinante di una matrice attraverso la matrice dei cofattori (o matrice dei complementi algebrici) di e della derivata di stessa. Il determinante di una matrice può infatti considerarsi una funzione polinomiale: quindi essa è differenziabile e il suo differenziale può essere espresso mediante la formula di Jacobi: dove denota la trasposta della matrice dei cofattori (detta anche matrice aggiunta e denotata come ), mentre è la traccia. Dunque la derivata rispetto a del determinante si scrive: (it)
  • Формула Якоби — формула, связывающая определитель матрицы, удовлетворяющей дифференциальному уравнению, в начале интервала интегрирования с определителем матрицы в конце интервала интегрирования. (ru)
  • 在矩阵微积分中，雅可比公式（Jacobi's formula）把矩阵 的行列式的导数表达为 的伴随矩阵与 本身导数的乘积的跡。 若 是从实数到 矩阵的可微映射，则 。 其中 為矩阵 的跡。 (zh)
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