Search results
Results From The WOW.Com Content Network
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix.The determinant of a matrix A is commonly denoted det(A), det A, or | A |.Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix.
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.
For example, the determinant of an n × n matrix is an SL(n) 2 invariant and Cayley's hyperdeterminant for a 2 × 2 × 2 hypermatrix is an SL(2) 3 invariant. A more familiar property of a determinant is that if you add a multiple of a row (or column) to a different row (or column) of a square matrix then its determinant is unchanged.
In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné ( 1943 ). If K is a division ring, then the Dieudonné determinant is a group homomorphism from the group GL n ( K ) of invertible n -by- n matrices over K onto the ...
If all singular values are zero, then the pseudo-determinant is 1. Supposing rank ( A ) = k {\displaystyle \operatorname {rank} (A)=k} , so that k is the number of non-zero singular values, we may write A = P P † {\displaystyle A=PP^{\dagger }} where P {\displaystyle P} is some n -by- k matrix and the dagger is the conjugate transpose .
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:
From this it is easily seen that all three matrices have determinant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for both L n and U n. In other words, matrices S n, L n, and U n are unimodular, with L n and U n having trace n. The trace of S n is given by = = [()]!