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For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo (the latter determinant ...
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant.
The determinant of a square Vandermonde matrix is called a Vandermonde polynomial or Vandermonde determinant.Its value is the polynomial = < ()which is non-zero if and only if all are distinct.
(In practical applications of numerical linear algebra, however, explicit computation of the determinant is rarely required.) See, for example, Trefethen & Bau (1997) . The determinant can also be evaluated in fewer than O ( n 3 ) {\displaystyle O(n^{3})} operations by reducing the problem to matrix multiplication , but most such algorithms are ...
Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals. In matrix theory , the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French ...
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a ...
The Hankel matrix transform, or simply Hankel transform, of a sequence is the sequence of the determinants of the Hankel matrices formed from .Given an integer >, define the corresponding ()-dimensional Hankel matrix as having the matrix elements [], = +.