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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
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 ≡ ...
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 last row of is the vector shifted by one in reverse. Different sources define the circulant matrix in different ways, for example as above, or with the vector c {\displaystyle c} corresponding to the first row rather than the first column of the matrix; and possibly with a different direction of shift (which is sometimes called an anti ...
In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrices and is defined as [1] [2] [3] = in which the ij-th block is the m i p i × n j q j sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal.
Note how the use of A[i][j] with multi-step indexing as in C, as opposed to a neutral notation like A(i,j) as in Fortran, almost inevitably implies row-major order for syntactic reasons, so to speak, because it can be rewritten as (A[i])[j], and the A[i] row part can even be assigned to an intermediate variable that is then indexed in a separate expression.
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case.