Transpose

File:Matrix transpose.gif

The transpose A^T 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 is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix Template:Math by producing another matrix, often denoted by Template:Math (among other notations).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.<ref>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.</ref>

Transpose of a matrix

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Definition

The transpose of a matrix Template:Math, denoted by Template:Math,<ref name="Whitelaw1991">Template:Cite book</ref> Template:Math, Template:Math, <math>A^{\intercal}</math>,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Math,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Math, Template:Math or Template:Math, may be constructed by any one of the following methods:

1. Reflect Template:Math over its main diagonal (which runs from top-left to bottom-right) to obtain Template:Math
2. Write the rows of Template:Math as the columns of Template:Math
3. Write the columns of Template:Math as the rows of Template:Math

Formally, the Template:Mvar-th row, Template:Mvar-th column element of Template:Math is the Template:Mvar-th row, Template:Mvar-th column element of Template:Math:

<math>\left[\mathbf{A}^\operatorname{T}\right]_{ij} = \left[\mathbf{A}\right]_{ji}.</math>

If Template:Math is an Template:Math matrix, then Template:Math is an Template:Math matrix.

In the case of square matrices, Template:Math may also denote the Template:Mathth power of the matrix Template:Math. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as Template:Math. An advantage of this notation is that no parentheses are needed when exponents are involved: as Template:Math, notation Template:Math is not ambiguous.

In this article, this confusion is avoided by never using the symbol Template:Math as a variable name.

Matrix definitions involving transposition

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

<math>\mathbf{A}^{\operatorname{T}} = \mathbf{A}.</math>

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

<math>\mathbf{A}^{\operatorname{T}} = -\mathbf{A}.</math>

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, Template:Math is Hermitian if

<math>\mathbf{A}^{\operatorname{T}} = \overline{\mathbf{A}}.</math>

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

<math>\mathbf{A}^{\operatorname{T}} = -\overline{\mathbf{A}}.</math>

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

<math>\mathbf{A}^{\operatorname{T}} = \mathbf{A}^{-1}.</math>

A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, Template:Math is unitary if

<math>\mathbf{A}^{\operatorname{T}} = \overline{\mathbf{A}^{-1}}.</math>

Examples

• <math>\begin{bmatrix}

   1 & 2
 \end{bmatrix}^{\operatorname{T}}
 = \,
 \begin{bmatrix}
   1   \\
   2
 \end{bmatrix}

</math>

• <math>

 \begin{bmatrix}
   1 & 2  \\
   3 & 4
 \end{bmatrix}^{\operatorname{T}}
 =
 \begin{bmatrix}
   1 & 3  \\
   2 & 4
 \end{bmatrix}

</math>

• <math>

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

</math>

Properties

Let Template:Math and Template:Math be matrices and Template:Mvar be a scalar.

• <math>\left(\mathbf{A}^\operatorname{T} \right)^\operatorname{T} = \mathbf{A}.</math>

  The operation of taking the transpose is an involution (self-inverse).

• <math>\left(\mathbf{A} + \mathbf{B}\right)^\operatorname{T} = \mathbf{A}^\operatorname{T} + \mathbf{B}^\operatorname{T}.</math>

  The transpose respects addition.

• <math>\left(c \mathbf{A}\right)^\operatorname{T} = c (\mathbf{A}^\operatorname{T}).</math>

  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 Template:Math matrices to the space of the Template:Math matrices.

• <math>\left(\mathbf{A B}\right)^\operatorname{T} = \mathbf{B}^\operatorname{T} \mathbf{A}^\operatorname{T}.</math>

  The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so

  Template:Math.

• <math>\det \left(\mathbf{A}^\operatorname{T}\right) = \det(\mathbf{A}).</math>

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

• The dot product of two column vectors Template:Math and Template:Math can be computed as the single entry of the matrix product<math display=block>\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^{\operatorname{T}} \mathbf{b}.</math>
• If Template:Math has only real entries, then Template:Math is a positive-semidefinite matrix.
• <math> \left(\mathbf{A}^\operatorname{T} \right)^{-1} = \left(\mathbf{A}^{-1} \right)^\operatorname{T}.</math>

  The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.
  The notation Template:Math is sometimes used to represent either of these equivalent expressions.

• If Template:Math is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose, since they share the same characteristic polynomial.
• <math> \left(\mathbf A\mathbf a\right) \cdot \mathbf b =\mathbf a \cdot \mathbf \left(A^T\mathbf b\right)</math> for two column vectors <math> \mathbf a, \mathbf b </math> and the standard dot product.
• Over any field <math>k</math>, a square matrix <math>\mathbf{A}</math> is similar to <math>\mathbf{A}^\operatorname{T}</math>.

  This implies that <math>\mathbf{A}</math> and <math>\mathbf{A}^\operatorname{T}</math> 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 <math>\mathbf{A}</math> and <math>\mathbf{B}</math> be <math>n\times n</math> matrices over some base field <math>k</math> and let <math>L</math> be a field extension of <math>k</math>. If <math>\mathbf{A}</math> and <math>\mathbf{B}</math> are similar as matrices over <math>L</math>, then they are similar over <math>k</math>. In particular this applies when <math>L</math> is the algebraic closure of <math>k</math>.
  • If <math>\mathbf{A}</math> is a matrix over an algebraically closed field in Jordan normal form with respect to some basis, then <math>\mathbf{A}</math> is similar to <math>\mathbf{A}^\operatorname{T}</math>. This further reduces to proving the same fact when <math>\mathbf{A}</math> is a single Jordan block, which is a straightforward exercise.

Products

If Template:Math is an Template:Math matrix and Template:Math is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: Template:Math is Template:Math and Template:Math is Template:Math. Furthermore, these products are symmetric matrices. Indeed, the matrix product Template:Math has entries that are the inner product of a row of Template:Math with a column of Template:Math. But the columns of Template:Math are the rows of Template:Math, so the entry corresponds to the inner product of two rows of Template:Math. If Template:Mvar is the entry of the product, it is obtained from rows Template:Mvar and Template:Mvar in Template:Math. The entry Template:Mvar is also obtained from these rows, thus Template:Math, and the product matrix (Template:Mvar) is symmetric. Similarly, the product Template:Math is a symmetric matrix.

A quick proof of the symmetry of Template:Math results from the fact that it is its own transpose:

<math>\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} .</math><ref>Gilbert Strang (2006) Linear Algebra and its Applications 4th edition, page 51, Thomson Brooks/Cole Template:ISBN</ref>

Implementation of matrix transposition on computers

File:Row and column major order.svg

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 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

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Let Template:Math denote the algebraic dual space of an Template:Mvar-module Template:Mvar. Let Template:Mvar and Template:Mvar be Template:Mvar-modules. If Template:Math is a linear map, then its algebraic adjoint or dual,Template:Sfn is the map Template:Math defined by Template:Math. The resulting functional Template:Math is called the pullback of Template:Mvar by Template:Mvar. The following relation characterizes the algebraic adjoint of Template:Mvar<ref>Template:Harvnb</ref>

Template:Math for all Template:Math and Template:Math

where Template:Math is the natural pairing (i.e. defined by Template:Math). This definition also applies unchanged to left modules and to vector spaces.<ref>Template:Harvnb</ref>

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) Template:Mvar is denoted by Template:Math. If Template:Mvar and Template:Mvar are TVSs then a linear map Template:Math is weakly continuous if and only if Template:Math, in which case we let Template:Math denote the restriction of Template:Math to Template:Math. The map Template:Math is called the transposeTemplate:Sfn of Template:Mvar.

If the matrix Template:Math describes a linear map with respect to bases of Template:Mvar and Template:Mvar, then the matrix Template:Math describes the transpose of that linear map with respect to the dual bases.

Transpose of a bilinear form

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Every linear map to the dual space Template:Math defines a bilinear form Template:Math, with the relation Template:Math. By defining the transpose of this bilinear form as the bilinear form Template:Mvar defined by the transpose Template:Math i.e. Template:Math, we find that Template:Math. Here, Template:Mvar is the natural homomorphism Template:Math into the double dual.

Adjoint

If the vector spaces Template:Mvar and Template:Mvar have respectively nondegenerate bilinear forms Template:Math and Template:Math, a concept known as the adjoint, which is closely related to the transpose, may be defined:

If Template:Nowrap is a linear map between vector spaces Template:Mvar and Template:Mvar, we define Template:Mvar as the adjoint of Template:Mvar if Template:Nowrap satisfies

<math>B_X\big(x, g(y)\big) = B_Y\big(u(x), y\big)</math> for all Template:Math and Template:Math.

These bilinear forms define an isomorphism between Template:Mvar and Template:Math, and between Template:Mvar and Template:Math, resulting in an isomorphism between the transpose and adjoint of Template:Mvar. 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 Template:Nowrap is equal to Template:Nowrap. In particular, this allows the orthogonal group over a vector space Template:Mvar with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps Template:Nowrap 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)

Further reading

• Template:Bourbaki Algebra I Chapters 1-3 Springer

• Template:Citation.
• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Cite book

External links

• Gilbert Strang (Spring 2010) Linear Algebra from MIT Open Courseware

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