# Dictionary Definition

transpose n : a matrix formed by interchanging
the rows and columns of a given matrix

### Verb

1 change the order or arrangement of; "Dyslexics
often transpose letters in a word" [syn: permute, commute]

2 transfer from one place or period to another;
"The ancient Greek story was transplanted into Modern America"
[syn: transfer,
transplant]

3 cause to change places; "interchange this screw
for one of a smaller size" [syn: counterchange, interchange]

4 transfer a quantity from one side of an
equation to the other side reversing its sign, in order to maintain
equality

5 put (a piece of music) into another key

6 transpose and remain equal in value; "These
operators commute with each other" [syn: commute]

7 change key; "Can you transpose this fugue into
G major?"

# User Contributed Dictionary

## English

### Etymology

From , from perfect passive participle transpositus, from transponere, to put across, from trans, across, and ponere, to put### Pronunciation

- Verb:
- or (UK)
- /trænzˈpəʊz/ or /trɑːnzˈpəʊz/ (UK)
- /tr

# 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

transpose in Catalan: Matriu
transposada

transpose in Czech:
Transpozice matice

transpose in Danish:
Transponering (matematik)

transpose in German: Matrix
(Mathematik)#Die_transponierte_Matrix

transpose in Esperanto:
Transpono

transpose in Estonian:
Transponeeritud maatriks

transpose in Spanish: Matriz
traspuesta

transpose in Basque: Matrize
irauli

transpose in French: Matrice
transposée

transpose in Korean:
전치행렬

transpose in Italian: Matrice
trasposta

transpose in Hebrew: מטריצה
משוחלפת

transpose in Dutch:
Getransponeerde matrix

transpose in Japanese:
転置行列

transpose in Polish: Macierz
transponowana

transpose in Portuguese:
Matriz transposta

transpose in Russian:
Транспонированная матрица

transpose in Finnish:
Transpoosi

transpose in Swedish: Matris
(matematik)#Transponat

transpose in Thai:
เมทริกซ์สลับเปลี่ยน

transpose in Vietnamese: Ma
trận chuyển vị

transpose in Ukrainian:
Транспонована матриця

transpose in Urdu: پلٹ
(میٹرکس)

transpose in Chinese:
转置矩阵

# Synonyms, Antonyms and Related Words

adapt, alternate, arrange, assign, bandy, be quits with, carry over,
change, communicate, commute, compensate, compose, consign, convert, cooperate, counterchange, deliver, deport, diffuse, disseminate, evert, exchange, expel, export, extradite, get back at, get
even with, give and take, hand forward, hand on, hand over,
harmonize, impart, import, instrument, instrumentate, interchange, introvert, intussuscept, invaginate, inverse, invert, logroll, make an adaptation,
make over, melodize,
metamorphose,
metastasize,
metathesize,
musicalize, orchestrate, pass, pass on, pass over, pass the
buck, pay back, perfuse,
permute, pronate, put, put to music, reciprocate, relay, render, requite, respond, resupinate, retaliate, return, return the compliment,
reverse, revert, revolve, rotate, score, set, set to music, spread, supinate, swap, switch, trade, transcribe, transfer, transfer property,
transfigure,
transfuse, translate, translocate, transmit, transmogrify, transmute, transplace, transplant, transubstantiate,
turn, turn about, turn
around, turn down, turn in, turn inside out, turn out, turn over,
turn the scale, turn the tables, turn upside down, write