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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 ...
B i consists of n block matrices of size m × m, stacked column-wise, and all these matrices are all-zero except for the i-th one, which is a m × m identity matrix I m. Then the vectorized version of X can be expressed as follows: vec ( X ) = ∑ i = 1 n B i X e i {\displaystyle \operatorname {vec} (\mathbf {X} )=\sum _{i=1}^{n}\mathbf {B ...
On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid data movement.
For cables, the swapping is gradual and continuous; that is the two or three conductors are twisted around each other. For communication cables, this is called twisted pair. For overhead power lines or open pair communication lines, the conductors are exchanged at pylons, for example at transposition towers or at utility poles, respectively.
For complex tables, when a header spans two columns or rows, use ! scope="colgroup" colspan="2" | or ! scope="rowgroup" rowspan="2" | respectively to clearly identify the header as a column header of two columns or a row header of two rows.
Double-circuit lines are usually set up with conductors of the same phase placed opposite each other. This reduces the reactance due to mutual inductance; the reactance of both circuits together is less than half that of one circuit. For example, a section of a line may be (top-to-bottom) phases A-B-C on the left, also phases C'-B'-A' on the right.
An elementary row operation is any one of the following moves: Swap: Swap two rows of a matrix. Scale: Multiply a row of a matrix by a nonzero constant. Pivot: Add a multiple of one row of a matrix to another row. Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations.
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.