World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0000173844
Reproduction Date:

Title: Transpose  
Author: World Heritage Encyclopedia
Language: English
Subject: Adjugate matrix, Square matrix, Dyadics, Linear algebra, Invertible matrix
Collection: Abstract Algebra, Linear Algebra, Matrices
Publisher: World Heritage Encyclopedia


This article is about the transpose of a matrix. For other uses, see Transposition
Note that this article assumes that matrices are taken over a commutative ring. These results may not hold in the non-commutative case.
The transpose AT of a matrix A can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.

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

  • reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT
  • write the rows of A as the columns of AT
  • write the columns of A as the rows of AT

Formally, the i th row, j th column element of AT is the j th row, i th column element of A:

[\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji}

If A is an m × n matrix then AT is an n × m matrix.

The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.[1]


  • Examples 1
  • Properties 2
  • Special transpose matrices 3
  • Transpose of a linear map 4
    • Transpose of a bilinear form 4.1
    • Adjoint 4.2
  • Implementation of matrix transposition on computers 5
  • See also 6
  • References 7
  • Further reading 8
  • External links 9


  • \begin{bmatrix} 1 & 2 \end{bmatrix}^{\mathrm{T}} = \, \begin{bmatrix} 1 \\ 2 \end{bmatrix}
  • \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}
  • \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix}


For matrices A, B and scalar c we have the following properties of transpose:

  1. ( \mathbf{A}^\mathrm{T} ) ^\mathrm{T} = \mathbf{A} \quad \,
    The operation of taking the transpose is an involution (self-inverse).
  2. (\mathbf{A}+\mathbf{B}) ^\mathrm{T} = \mathbf{A}^\mathrm{T} + \mathbf{B}^\mathrm{T} \,
    The transpose respects addition.
  3. \left( \mathbf{A B} \right) ^\mathrm{T} = \mathbf{B}^\mathrm{T} \mathbf{A}^\mathrm{T} \,
    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. By induction this result extends to the general case of multiple matrices, where we find that (A1A2...Ak−1Ak)T = AkTAk−1T...A2TA1T.
  4. (c \mathbf{A})^\mathrm{T} = c \mathbf{A}^\mathrm{T} \,
    The transpose of a scalar is the same scalar. Together with (2), this states that the transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.
  5. \det(\mathbf{A}^\mathrm{T}) = \det(\mathbf{A}) \,
    The determinant of a square matrix is the same as that of its transpose.
  6. The dot product of two column vectors a and b can be computed as the single entry of the matrix product
    \left[ \mathbf{a} \cdot \mathbf{b} \right] = \mathbf{a}^{\mathrm{T}} \mathbf{b} ,
    which is written as aibi in Einstein summation convention.
  7. If A has only real entries, then ATA is a positive-semidefinite matrix.
  8. (\mathbf{A}^\mathrm{T})^{-1} = (\mathbf{A}^{-1})^\mathrm{T} \,
    The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A−T is sometimes used to represent either of these equivalent expressions.
  9. If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose since they share the same characteristic polynomial.
  10. \mathbf{A} \mathbf{A}^\mathrm{T} is a symmetric matrix.[2]

Special transpose matrices

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if

\mathbf{A}^{\mathrm{T}} = \mathbf{A} .

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

\mathbf{A}^{\mathrm{T}} = -\mathbf{A} .

A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if

\mathbf{A}^{\mathrm{T}} = \overline{\mathbf{A}} .

A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if

\mathbf{A}^{\mathrm{T}} = -\overline{\mathbf{A}} .

A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if

\mathbf{A}^{\mathrm{T}} = \mathbf{A}^{-1} .

Transpose of a linear map

The transpose may be defined in a more generally:

If f : VW is a linear map between right R-modules V and W with respective dual modules V and W, the transpose of f is the linear map

{}^\mathrm{t} f : W^* \to V^* : \varphi \mapsto \varphi \circ f .

Equivalently, the transpose tf is defined by the relation

\left\langle \phi , f ( v ) \right\rangle = \left\langle {}^\mathrm{t} f ( \phi ) , v \right\rangle \quad \forall \phi \in W^* , v \in V ,

where ·,· is the duality pairing. This definition also applies unchanged to left modules and to vector spaces.[3]

The definition of the transpose may be seen to be independent of any bilinear form on the vector spaces, unlike the adjoint (below).

If the matrix A describes a linear map with respect to bases of V and W, then the matrix AT describes the transpose of that linear map with respect to the dual bases.

Transpose of a bilinear form

Every linear map to the dual space f : VV defines a bilinear form B : V × VF, with the relation B(v, w) = f(v)(w). By defining the transpose of this bilinear form as the bilinear form tB defined by the transpose tf : V∗∗V i.e. tB(w, v) = tf(w)(v), we find that B(v, w) = tB(w, v).


If the vector spaces V and W have respective nondegenerate bilinear forms BV and BW, a concept closely related to the transpose – the adjoint – may be defined:

If f : VW is a linear map between vector spaces V and W, we define g as the adjoint of f if g : WV satisfies

B_V(v, g(w)) = B_W(f(v),w) \quad \forall v \in V, w \in W .

These bilinear forms define an isomorphism between V and V, and between W and W, resulting in an isomorphism between the transpose and adjoint of f. The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors use the term transpose to refer to the adjoint as defined here.

The adjoint allows us to consider whether g : WV is equal to f −1 : WV. In particular, this allows the orthogonal group over a vector space V with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps VV for which the adjoint equals the inverse.

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

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.

See also


  1. ^ Arthur Cayley (1858) "A memoir on the theory of matrices," Philosophical Transactions of the Royal Society of London, 148 : 17-37. The transpose (or "transposition") is defined on page 31.
  2. ^ "Is a matrix multiplied with its transpose something special?" Proof in answer on,
  3. ^ Bourbaki, "II §2.5", Algebra I 

Further reading

  • Maruskin, Jared M. (2012). Essential Linear Algebra. San José: Solar Crest. pp. 122–132.  
  • Schwartz, Jacob T. (2001). Introduction to Matrices and Vectors. Mineola: Dover. pp. 126–132.  

External links

  • MIT Linear Algebra Lecture on Matrix Transposes
  • Transpose,
  • Transpose,
  • Khan Academy introduction to matrix transposes
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.