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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]
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
To add an extra row into a table, you'll need to insert an extra row break and the same number of new cells as are in the other rows. The easiest way to do this in practice, is to duplicate an existing row by copying and pasting the markup. It's then just a matter of editing the cell contents.
Similarly, a row vector is a matrix for some , consisting of a single row of entries, = […]. (Throughout this article, boldface is used for both row and column vectors.) 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 [] = […].
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.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
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 ≡ ...
The matrix B of a bilinear form B on a basis (, …,) (the "old" basis in what follows) is the matrix whose entry of the i th row and j th column is (,). It follows that if v and w are the column vectors of the coordinates of two vectors v and w, one has