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Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2 n−1 times the maximal determinant of a {0,1} matrix of size n−1.
There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending ...
Download as PDF; Printable version; ... Pages in category "Matrices" The following 200 pages are in this category, out of approximately 235 total. ... Hadamard matrix ...
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:
Download as PDF; Printable version; ... Pages in category "Determinants" The following 47 pages are in this category, out of 47 total. ... Jacobian matrix and ...
where is the matrix formed by replacing the i-th column of A by the column vector b. A more general version of Cramer's rule [13] considers the matrix equation = where the n × n matrix A has a nonzero determinant, and X, B are n × m matrices.
The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one. The determinant of a square matrix A (denoted det(A) or | A |) is a number encoding
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...