Search results
Results from the WOW.Com Content Network
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.
The mortal matrix problem. ... For example, there is an ... Rule 110 - most questions involving "can property X appear later" are undecidable.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A is denoted A −1, and, thus verifies = =. A matrix that has an inverse is an invertible matrix.
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 ...
For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [10] for a total of approximately 2n 3 /3 operations.
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.
A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is