
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).

The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.
Transpose of a matrix
Definition
The transpose of a matrix A, denoted by AT,⊤A, A⊤, ,A′,Atr, tA or At, may be constructed by any one of the following methods:
- 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:
If A is an m × n matrix, then AT is an n × m matrix.
In the case of square matrices, AT may also denote the Tth power of the matrix A. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as TA. An advantage of this notation is that no parentheses are needed when exponents are involved: as (TA)n = T(An), notation TAn is not ambiguous.
In this article, this confusion is avoided by never using the symbol T as a variable name.
Matrix definitions involving transposition
A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if
A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if
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
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
A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if
A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if
Examples
Properties
Let A and B be matrices and c be a scalar.
- The operation of taking the transpose is an involution (self-inverse).
- The transpose respects addition.
- The transpose of a scalar is the same scalar. Together with the preceding property, this implies that the transpose is a linear map from the space of m × n matrices to the space of the n × m matrices.
- The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so
- (A1A2...Ak−1Ak)T = AkTAk−1T…A2TA1T.
- The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so
- The determinant of a square matrix is the same as the determinant of its transpose.
- The dot product of two column vectors a and b can be computed as the single entry of the matrix product
- If A has only real entries, then ATA is a positive-semidefinite matrix.
- 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.
- The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.
- If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose, since they share the same characteristic polynomial.
for two column vectors
and the standard dot product.
- Over any field
, a square matrix
is similar to
.
- This implies that
and
have the same invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties.
- A proof of this property uses the following two observations.
- Let
and
be
matrices over some base field
and let
be a field extension of
. If
and
are similar as matrices over
, then they are similar over
. In particular this applies when
is the algebraic closure of
.
- If
is a matrix over an algebraically closed field in Jordan normal form with respect to some basis, then
is similar to
. This further reduces to proving the same fact when
is a single Jordan block, which is a straightforward exercise.
- Let
- This implies that
Products
If A is an m × n matrix and AT is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A AT is m × m and ATA is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A AT has entries that are the inner product of a row of A with a column of AT. But the columns of AT are the rows of A, so the entry corresponds to the inner product of two rows of A. If pi j is the entry of the product, it is obtained from rows i and j in A. The entry pj i is also obtained from these rows, thus pi j = pj i, and the product matrix (pi j) is symmetric. Similarly, the product ATA is a symmetric matrix.
A quick proof of the symmetry of A AT results from the fact that it is its own transpose:
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.
Transposes of linear maps and bilinear forms
As the main use of matrices is to represent linear maps between finite-dimensional vector spaces, the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps.
This leads to a much more general definition of the transpose that works on every linear map, even when linear maps cannot be represented by matrices (such as in the case of infinite dimensional vector spaces). In the finite dimensional case, the matrix representing the transpose of a linear map is the transpose of the matrix representing the linear map, independently of the basis choice.
Transpose of a linear map
Let X# denote the algebraic dual space of an R-module X. Let X and Y be R-modules. If u : X → Y is a linear map, then its algebraic adjoint or dual, is the map u# : Y# → X# defined by f ↦ f ∘ u. The resulting functional u#(f) is called the pullback of f by u. The following relation characterizes the algebraic adjoint of u
- ⟨u#(f), x⟩ = ⟨f, u(x)⟩ for all f ∈ Y# and x ∈ X
where ⟨•, •⟩ is the natural pairing (i.e. defined by ⟨h, z⟩ := h(z)). This definition also applies unchanged to left modules and to vector spaces.
The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint (below).
The continuous dual space of a topological vector space (TVS) X is denoted by X'. If X and Y are TVSs then a linear map u : X → Y is weakly continuous if and only if u#(Y') ⊆ X', in which case we let tu : Y' → X' denote the restriction of u# to Y'. The map tu is called the transpose of u.
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 u : X → X# defines a bilinear form B : X × X → F, with the relation B(x, y) = u(x)(y). By defining the transpose of this bilinear form as the bilinear form tB defined by the transpose tu : X## → X# i.e. tB(y, x) = tu(Ψ(y))(x), we find that B(x, y) = tB(y, x). Here, Ψ is the natural homomorphism X → X## into the double dual.
Adjoint
If the vector spaces X and Y have respectively nondegenerate bilinear forms BX and BY, a concept known as the adjoint, which is closely related to the transpose, may be defined:
If u : X → Y is a linear map between vector spaces X and Y, we define g as the adjoint of u if g : Y → X satisfies
for all x ∈ X and y ∈ Y.
These bilinear forms define an isomorphism between X and X#, and between Y and Y#, resulting in an isomorphism between the transpose and adjoint of u. 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 however, use the term transpose to refer to the adjoint as defined here.
The adjoint allows us to consider whether g : Y → X is equal to u −1 : Y → X. In particular, this allows the orthogonal group over a vector space X with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps X → X 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.
See also
- Adjugate matrix, the transpose of the cofactor matrix
- Conjugate transpose
- Converse relation
- Moore–Penrose pseudoinverse
- Projection (linear algebra)
References
- Nykamp, Duane. "The transpose of a matrix". Math Insight. Retrieved September 8, 2020.
- 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.
- T.A. Whitelaw (1 April 1991). Introduction to Linear Algebra, 2nd edition. CRC Press. ISBN 978-0-7514-0159-2.
- "Transpose of a Matrix Product (ProofWiki)". ProofWiki. Retrieved 4 Feb 2021.
- "What is the best symbol for vector/matrix transpose?". Stack Exchange. Retrieved 4 Feb 2021.
- Weisstein, Eric W. "Transpose". mathworld.wolfram.com. Retrieved 2020-09-08.
- Gilbert Strang (2006) Linear Algebra and its Applications 4th edition, page 51, Thomson Brooks/Cole ISBN 0-03-010567-6
- Schaefer & Wolff 1999, p. 128.
- Halmos 1974, §44
- Bourbaki 1989, II §2.5
- Trèves 2006, p. 240.
Further reading
- Bourbaki, Nicolas (1989) [1970]. Algebra I Chapters 1-3 [Algèbre: Chapitres 1 à 3] (PDF). Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64243-5. OCLC 18588156.
- Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3.
- Maruskin, Jared M. (2012). Essential Linear Algebra. San José: Solar Crest. pp. 122–132. ISBN 978-0-9850627-3-6.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Schwartz, Jacob T. (2001). Introduction to Matrices and Vectors. Mineola: Dover. pp. 126–132. ISBN 0-486-42000-0.
External links
- Gilbert Strang (Spring 2010) Linear Algebra from MIT Open Courseware
In linear algebra the transpose of a matrix is an operator which flips a matrix over its diagonal that is it switches the row and column indices of the matrix A by producing another matrix often denoted by AT among other notations 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 The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley Transpose of a matrixDefinition The transpose of a matrix A denoted by AT A A A displaystyle A intercal A Atr tA or At may be constructed by any one of the following methods 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 AT ij A ji displaystyle left mathbf A operatorname T right ij left mathbf A right ji If A is an m n matrix then AT is an n m matrix In the case of square matrices AT may also denote the T th power of the matrix A For avoiding a possible confusion many authors use left upperscripts that is they denote the transpose as TA An advantage of this notation is that no parentheses are needed when exponents are involved as TA n T An notation TAn is not ambiguous In this article this confusion is avoided by never using the symbol T as a variable name Matrix definitions involving transposition A square matrix whose transpose is equal to itself is called a symmetric matrix that is A is symmetric if AT A displaystyle mathbf A operatorname 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 AT A displaystyle mathbf A operatorname 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 AT A displaystyle mathbf A operatorname 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 AT A displaystyle mathbf A operatorname 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 AT A 1 displaystyle mathbf A operatorname T mathbf A 1 A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix that is A is unitary if AT A 1 displaystyle mathbf A operatorname T overline mathbf A 1 Examples 12 T 12 displaystyle begin bmatrix 1 amp 2 end bmatrix operatorname T begin bmatrix 1 2 end bmatrix 1234 T 1324 displaystyle begin bmatrix 1 amp 2 3 amp 4 end bmatrix operatorname T begin bmatrix 1 amp 3 2 amp 4 end bmatrix 123456 T 135246 displaystyle begin bmatrix 1 amp 2 3 amp 4 5 amp 6 end bmatrix operatorname T begin bmatrix 1 amp 3 amp 5 2 amp 4 amp 6 end bmatrix Properties Let A and B be matrices and c be a scalar AT T A displaystyle left mathbf A operatorname T right operatorname T mathbf A The operation of taking the transpose is an involution self inverse A B T AT BT displaystyle left mathbf A mathbf B right operatorname T mathbf A operatorname T mathbf B operatorname T The transpose respects addition cA T c AT displaystyle left c mathbf A right operatorname T c mathbf A operatorname T The transpose of a scalar is the same scalar Together with the preceding property this implies that the transpose is a linear map from the space of m n matrices to the space of the n m matrices AB T BTAT displaystyle left mathbf AB right operatorname T mathbf B operatorname T mathbf A operatorname T The order of the factors reverses By induction this result extends to the general case of multiple matrices so A1A2 Ak 1Ak T AkTAk 1T A2TA1T dd det AT det A displaystyle det left mathbf A operatorname T right det mathbf A The determinant of a square matrix is the same as the determinant of its transpose The dot product of two column vectors a and b can be computed as the single entry of the matrix producta b aTb displaystyle mathbf a cdot mathbf b mathbf a operatorname T mathbf b If A has only real entries then ATA is a positive semidefinite matrix AT 1 A 1 T displaystyle left mathbf A operatorname T right 1 left mathbf A 1 right operatorname 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 If A is a square matrix then its eigenvalues are equal to the eigenvalues of its transpose since they share the same characteristic polynomial Aa b a ATb displaystyle left mathbf A mathbf a right cdot mathbf b mathbf a cdot mathbf left A T mathbf b right for two column vectors a b displaystyle mathbf a mathbf b and the standard dot product Over any field k displaystyle k a square matrix A displaystyle mathbf A is similar to AT displaystyle mathbf A operatorname T This implies that A displaystyle mathbf A and AT displaystyle mathbf A operatorname T have the same invariant factors which implies they share the same minimal polynomial characteristic polynomial and eigenvalues among other properties A proof of this property uses the following two observations Let A displaystyle mathbf A and B displaystyle mathbf B be n n displaystyle n times n matrices over some base field k displaystyle k and let L displaystyle L be a field extension of k displaystyle k If A displaystyle mathbf A and B displaystyle mathbf B are similar as matrices over L displaystyle L then they are similar over k displaystyle k In particular this applies when L displaystyle L is the algebraic closure of k displaystyle k If A displaystyle mathbf A is a matrix over an algebraically closed field in Jordan normal form with respect to some basis then A displaystyle mathbf A is similar to AT displaystyle mathbf A operatorname T This further reduces to proving the same fact when A displaystyle mathbf A is a single Jordan block which is a straightforward exercise Products If A is an m n matrix and AT is its transpose then the result of matrix multiplication with these two matrices gives two square matrices A AT is m m and ATA is n n Furthermore these products are symmetric matrices Indeed the matrix product A AT has entries that are the inner product of a row of A with a column of AT But the columns of AT are the rows of A so the entry corresponds to the inner product of two rows of A If pi j is the entry of the product it is obtained from rows i and j in A The entry pj i is also obtained from these rows thus pi j pj i and the product matrix pi j is symmetric Similarly the product ATA is a symmetric matrix A quick proof of the symmetry of A AT results from the fact that it is its own transpose AAT T AT TAT AAT displaystyle left mathbf A mathbf A operatorname T right operatorname T left mathbf A operatorname T right operatorname T mathbf A operatorname T mathbf A mathbf A operatorname T Implementation of matrix transposition on computers Illustration of row and column major order 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 Transposes of linear maps and bilinear formsAs the main use of matrices is to represent linear maps between finite dimensional vector spaces the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps This leads to a much more general definition of the transpose that works on every linear map even when linear maps cannot be represented by matrices such as in the case of infinite dimensional vector spaces In the finite dimensional case the matrix representing the transpose of a linear map is the transpose of the matrix representing the linear map independently of the basis choice Transpose of a linear map Let X denote the algebraic dual space of an R module X Let X and Y be R modules If u X Y is a linear map then its algebraic adjoint or dual is the map u Y X defined by f f u The resulting functional u f is called the pullback of f by u The following relation characterizes the algebraic adjoint of u u f x f u x for all f Y and x X where is the natural pairing i e defined by h z h z This definition also applies unchanged to left modules and to vector spaces The definition of the transpose may be seen to be independent of any bilinear form on the modules unlike the adjoint below The continuous dual space of a topological vector space TVS X is denoted by X If X and Y are TVSs then a linear map u X Y is weakly continuous if and only if u Y X in which case we let tu Y X denote the restriction of u to Y The map tu is called the transpose of u 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 u X X defines a bilinear form B X X F with the relation B x y u x y By defining the transpose of this bilinear form as the bilinear form tB defined by the transpose tu X X i e tB y x tu PS y x we find that B x y tB y x Here PS is the natural homomorphism X X into the double dual Adjoint If the vector spaces X and Y have respectively nondegenerate bilinear forms BX and BY a concept known as the adjoint which is closely related to the transpose may be defined If u X Y is a linear map between vector spaces X and Y we define g as the adjoint of u if g Y X satisfies BX x g y BY u x y displaystyle B X big x g y big B Y big u x y big for all x X and y Y These bilinear forms define an isomorphism between X and X and between Y and Y resulting in an isomorphism between the transpose and adjoint of u 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 however use the term transpose to refer to the adjoint as defined here The adjoint allows us to consider whether g Y X is equal to u 1 Y X In particular this allows the orthogonal group over a vector space X with a quadratic form to be defined without reference to matrices nor the components thereof as the set of all linear maps X X 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 See alsoAdjugate matrix the transpose of the cofactor matrix Conjugate transpose Converse relation Moore Penrose pseudoinverse Projection linear algebra ReferencesNykamp Duane The transpose of a matrix Math Insight Retrieved September 8 2020 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 T A Whitelaw 1 April 1991 Introduction to Linear Algebra 2nd edition CRC Press ISBN 978 0 7514 0159 2 Transpose of a Matrix Product ProofWiki ProofWiki Retrieved 4 Feb 2021 What is the best symbol for vector matrix transpose Stack Exchange Retrieved 4 Feb 2021 Weisstein Eric W Transpose mathworld wolfram com Retrieved 2020 09 08 Gilbert Strang 2006 Linear Algebra and its Applications 4th edition page 51 Thomson Brooks Cole ISBN 0 03 010567 6 Schaefer amp Wolff 1999 p 128 Halmos 1974 44 Bourbaki 1989 II 2 5 Treves 2006 p 240 Further readingBourbaki Nicolas 1989 1970 Algebra I Chapters 1 3 Algebre Chapitres 1 a 3 PDF Elements de mathematique Berlin New York Springer Science amp Business Media ISBN 978 3 540 64243 5 OCLC 18588156 Halmos Paul 1974 Finite dimensional vector spaces Springer ISBN 978 0 387 90093 3 Maruskin Jared M 2012 Essential Linear Algebra San Jose Solar Crest pp 122 132 ISBN 978 0 9850627 3 6 Schaefer Helmut H Wolff Manfred P 1999 Topological Vector Spaces GTM Vol 8 Second ed New York NY Springer New York Imprint Springer ISBN 978 1 4612 7155 0 OCLC 840278135 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 Schwartz Jacob T 2001 Introduction to Matrices and Vectors Mineola Dover pp 126 132 ISBN 0 486 42000 0 External linksGilbert Strang Spring 2010 Linear Algebra from MIT Open Courseware