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Similarly, vec(A T) is the vector obtaining by vectorizing A in row-major order. The cycles and other properties of this permutation have been heavily studied for in-place matrix transposition algorithms. In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator [1]
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
Programming languages that implement matrices may have easy means for vectorization. In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well.
Some compiled languages such as Ada and Fortran, and some scripting languages such as IDL, MATLAB, and S-Lang, have native support for vectorized operations on arrays. For example, to perform an element by element sum of two arrays, a and b to produce a third c , it is only necessary to write
Typically, the matrix is assumed to be stored in row-major or column-major order (i.e., contiguous rows or columns, respectively, arranged consecutively). Performing an in-place transpose (in-situ transpose) is most difficult when N ≠ M , i.e. for a non-square (rectangular) matrix, where it involves a complex permutation of the data elements ...
Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...
The MATLAB language introduces the left-division operator \ to maintain the essential part of the analogy with the scalar case, therefore simplifying the mathematical reasoning and preserving the conciseness: A \ (A * x)==A \ b (A \ A)* x ==A \ b (associativity also holds for matrices, commutativity is no more required) x = A \ b