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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 ...
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
new row To begin a new row of cells, use a single vertical bar (|) and a hyphen (-). | new cell in row To add a new cell in a row, start each new cell with a new line and a single vertical bar (|), or several cells can be placed consecutively on the same line, separated by double vertical bars (||). |} end
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, = for matrices A, B, and C of dimensions k×l, l×m, and m×n.
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 ...
The transpose of a matrix A, denoted by A T, [3] ⊤ A, A ⊤, , [4] [5] A′, [6] A tr, t A or A t, may be constructed by any one of the following methods: Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain A T; Write the rows of A as the columns of A T; Write the columns of A as the rows of A T
There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent .
If is a linear transformation mapping to and is a column vector with entries, then = for some matrix , called the transformation matrix of . [ citation needed ] Note that A {\displaystyle A} has m {\displaystyle m} rows and n {\displaystyle n} columns, whereas the transformation T {\displaystyle T} is from R n {\displaystyle \mathbb {R} ^{n ...