AskDefine | Define transpose

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
1 & 2 \\ 3 & 4 \end^ \!\! \;\! = \, \begin 1 & 3 \\ 2 & 4 \end.
\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).
(\mathbf+\mathbf) ^\mathrm = \mathbf^\mathrm + \mathbf^\mathrm \,
The transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.
\left( \mathbf \right) ^\mathrm = \mathbf^\mathrm \mathbf^\mathrm \,
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.
(c \mathbf)^\mathrm = c \mathbf^\mathrm \,
The transpose of a scalar is the same scalar.
\det(\mathbf^\mathrm) = \det(\mathbf) \,
The determinant of a matrix is the same as that of its transpose.
The dot product of two column vectors a and b can be computed as
\mathbf \cdot \mathbf = \mathbf^ \mathbf,
which is written as ai bi in Einstein notation. If A has only real entries, then ATA is a positive-semidefinite matrix. If A is over some field, then A is similar to AT. (\mathbf^\mathrm)^ = (\mathbf^)^\mathrm \,
For an invertible matrix A, the transpose of the inverse is the inverse of the transpose.
If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its 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.
Here, BV and BW are the bilinear forms on V and W respectively. The matrix of the transpose of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms.
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

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
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