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This property is the reason that this matrix is referred to as the "swap operator" in the context of quantum information theory. Two explicit forms for the commutation matrix are as follows: if e r , j denotes the j -th canonical vector of dimension r (i.e. the vector with 1 in the j -th coordinate and 0 elsewhere) then
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.
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 ...
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 conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
Each column in an SQL table declares the type(s) that column may contain. ANSI SQL includes the following data types. [14] Character strings and national character strings. CHARACTER(n) (or CHAR(n)): fixed-width n-character string, padded with spaces as needed; CHARACTER VARYING(n) (or VARCHAR(n)): variable-width string with a maximum size of n ...
MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities.
Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa. As a linear transformation, a lower shift matrix shifts the components of a column vector one position down, with a zero appearing in the first position. An upper shift matrix shifts the components of a column vector one position up, with a zero appearing in ...