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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.
The matrix vectorization operation can be written in terms of a linear sum. Let X be an m × n matrix that we want to vectorize, and let e i be the i -th canonical basis vector for the n -dimensional space, that is e i = [ 0 , … , 0 , 1 , 0 , … , 0 ] T {\textstyle \mathbf {e} _{i}=\left[0,\dots ,0,1,0,\dots ,0\right]^{\mathrm {T} }} .
Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function f defined on a set D ⊂ C, one has = , where C is a closed simple curve inside the domain D enclosing x.
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite ...
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 ...
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Let Ω(n,k) be the class of all (0, 1)-matrices of order n with each row and column sum equal to k. Every matrix A in this class has perm(A) > 0. [13] The incidence matrices of projective planes are in the class Ω(n 2 + n + 1, n + 1) for n an integer > 1. The permanents corresponding to the smallest projective planes have been calculated.