# Dictionary Definition

# User Contributed Dictionary

## English

### Verb

transposed- past of transpose

# Extensive Definition

otheruses transposition

In linear
algebra, the transpose of a matrix
A is another matrix AT (also written Atr, tA, or A′)
created by any one of the following equivalent actions:

- write the rows of A as the columns of AT
- write the columns of A as the rows of AT
- reflect A by its main diagonal (which starts from the top left) to obtain AT

Formally, the transpose of an m × n matrix A is
the n × m matrix

- \mathbf^\mathrm_ = \mathbf_ for 1 \le i \le n, 1 \le j \le m.

## Examples

- \begin

\begin 1 & 2 \\ 3 & 4 \\ 5 & 6 \end^
\!\! \;\! = \, \begin 1 & 3 & 5\\ 2 & 4 & 6 \end.
\;

## Properties

For matrices A, B and scalar c we have the following properties of transpose:\left( \mathbf^\mathrm \right) ^\mathrm = \mathbf
\quad \,

- Taking the transpose is an involution (self inverse).

- The transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.

- Note that the order of the factors reverses. From this one can deduce that a square matrix A is invertible if and only if AT is invertible, and in this case we have (A−1)T = (AT)−1. It is relatively easy to extend this result to the general case of multiple matrices, where we find that (ABC...XYZ)T = ZTYTXT...CTBTAT.

- The transpose of a scalar is the same scalar.

- The determinant of a matrix is the same as that of its transpose.

- \mathbf \cdot \mathbf = \mathbf^ \mathbf,

- For an invertible matrix A, the transpose of the inverse is the inverse of the transpose.

## Special transpose matrices

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if- \mathbf^ = \mathbf.\,

A square matrix whose transpose is also its
inverse is called an orthogonal
matrix; that is, G is orthogonal if

- \mathbf^\mathrm = \mathbf^\mathrm \mathbf = \mathbf_n , \, the identity matrix, i.e. GT = G-1.

A square matrix whose transpose is equal to its
negative is called skew-symmetric
matrix; that is, A is skew-symmetric if

- \mathbf^ = -\mathbf.\,

The conjugate
transpose of the complex
matrix A, written as A*, is obtained by taking the transpose of A
and the complex
conjugate of each entry:

- \mathbf^* = (\overline)^ = \overline.

## Transpose of linear maps

If f: V→W is a linear map between vector spaces V and W with nondegenerate bilinear forms, we define the transpose of f to be the linear map tf : W→V, determined by- B_V(v,^tf(w))=B_W(f(v),w) \quad \forall\ v \in V, w \in W.

Over a complex vector space, one often works with
sesquilinear
forms instead of bilinear (conjugate-linear in one argument).
The transpose of a map between such spaces is defined similarly,
and the matrix of the transpose map is given by the conjugate
transpose matrix if the bases are orthonormal. In this case, the
transpose is also called the Hermitian
adjoint.

If V and W do not have bilinear forms, then the
transpose of a linear map f: V→W is only defined as a
linear map tf : W*→V* between the dual spaces of
W and V.

## Implementation of matrix transposition on computers

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 the necessity of data
movement.

However, there remain a number of circumstances
in which it is necessary or desirable to physically reorder a
matrix in memory to its transposed ordering. For example, with a
matrix stored in row-major
order, the rows of the matrix are contiguous in memory and the
columns are discontiguous. If repeated operations need to be
performed on the columns, for example in a fast
Fourier transform algorithm, transposing the matrix in memory
(to make the columns contiguous) may improve performance by
increasing memory
locality.

Ideally, one might hope to transpose a matrix
with minimal additional storage. This leads to the problem of
transposing an N × M matrix in-place, with
O(1) additional storage or at most storage much less than MN. For
N ≠ M, this involves a complicated
permutation of the
data elements that is non-trivial to implement in-place. Therefore
efficient
in-place matrix transposition has been the subject of numerous
research publications in computer
science, starting in the late 1950s, and several
algorithms have been developed.

## External links

- MIT Video Lecture on Matrix Transposes at Google Video, from MIT OpenCourseWare
- Transpose, mathworld.wolfram.com
- Transpose, planetmath.org

transposed in Catalan: Matriu transposada

transposed in Czech: Transpozice matice

transposed in Danish: Transponering
(matematik)

transposed in German: Matrix
(Mathematik)#Die_transponierte_Matrix

transposed in Esperanto: Transpono

transposed in Estonian: Transponeeritud
maatriks

transposed in Spanish: Matriz traspuesta

transposed in Basque: Matrize irauli

transposed in French: Matrice transposée

transposed in Korean: 전치행렬

transposed in Italian: Matrice trasposta

transposed in Hebrew: מטריצה משוחלפת

transposed in Dutch: Getransponeerde
matrix

transposed in Japanese: 転置行列

transposed in Polish: Macierz
transponowana

transposed in Portuguese: Matriz
transposta

transposed in Russian: Транспонированная
матрица

transposed in Finnish: Transpoosi

transposed in Swedish: Matris
(matematik)#Transponat

transposed in Thai: เมทริกซ์สลับเปลี่ยน

transposed in Vietnamese: Ma trận chuyển
vị

transposed in Ukrainian: Транспонована
матриця

transposed in Urdu: پلٹ (میٹرکس)

transposed in Chinese: 转置矩阵

# Synonyms, Antonyms and Related Words

arsy-varsy, ass over elbows, back-to-front,
backwards, capsized, changeable, chiastic, commutable, commutative, convertible, equal, equalizing, equivalent, even, everted, exchanged, give-and-take,
hyperbatic, inside
out, interchangeable,
interchanged,
introverted,
invaginated,
inversed, inverted, mutual, outside in, palindromic, permutable, reciprocal, reciprocating, reciprocative, resupinate, retaliatory, retroverted, returnable, reversed, standard, swapped, switched, topsy-turvy, traded, upside-down, wrong side
out