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Linear Algebra

from MIT Open Courseware {{Tensors Matrices Abstract algebra Linear algebra

linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, 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 by producing another matrix, often denoted by (among other notations).
The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians ...

. In the case of a logical matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation
In mathematics, a binary relation associates ele ...

representing a binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

R, the transpose corresponds to the converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...

RTranspose of a matrix

Definition

The transpose of a matrix , denoted by , , , $A^$, , , or , may be constructed by any one of the following methods: # Reflect over itsmain diagonal
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

(which runs from top-left to bottom-right) to obtain
#Write the rows of as the columns of
#Write the columns of as the rows of
Formally, the -th row, -th column element of is the -th row, -th column element of :
:$\backslash left;\; href="/html/ALL/s/mathbf^\backslash operatorname\backslash right.html"\; ;"title="mathbf^\backslash operatorname\backslash right">mathbf^\backslash operatorname\backslash right$
If is an matrix, then is an matrix.
In the case of square matrices, may also denote the th power of the matrix . For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as . An advantage of this notation is that no parentheses are needed when exponents are involved: as , notation is not ambiguous.
In this article this confusion is avoided by never using the symbol as a variable name.
Matrix definitions involving transposition

A square matrix whose transpose is equal to itself is called a ''symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...

''; that is, is symmetric if
:$\backslash mathbf^\; =\; \backslash mathbf.$
A square matrix whose transpose is equal to its negative is called a ''skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...

''; that is, is skew-symmetric if
:$\backslash mathbf^\; =\; -\backslash mathbf.$
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

(denoted here with an overline) is called a ''Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex number, complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the ele ...

'' (equivalent to the matrix being equal to its conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n Complex number, complex matrix (mathematics), matrix \boldsymbol is an n \times m matrix obtained by transpose, transposing \boldsymbol and applying ...

); that is, is Hermitian if
:$\backslash mathbf^\; =\; \backslash overline.$
A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a ''skew-Hermitian matrix
__NOTOC__
In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...

''; that is, is skew-Hermitian if
:$\backslash mathbf^\; =\; -\backslash overline.$
A square matrix whose transpose is equal to its inverse is called an ''orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are Orthonormality, orthonormal Vector (mathematics and physics), vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
wh ...

''; that is, is orthogonal if
:$\backslash mathbf^\; =\; \backslash mathbf^.$
A square complex matrix whose transpose is equal to its conjugate inverse is called a ''unitary matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

''; that is, is unitary if
:$\backslash mathbf^\; =\; \backslash overline.$
Examples

*$\backslash begin\; 1\; \&\; 2\; \backslash end^\; =\; \backslash ,\; \backslash begin\; 1\; \backslash \backslash \; 2\; \backslash end$ *$\backslash begin\; 1\; \&\; 2\; \backslash \backslash \; 3\; \&\; 4\; \backslash end^\; =\; \backslash begin\; 1\; \&\; 3\; \backslash \backslash \; 2\; \&\; 4\; \backslash end$ * $\backslash begin\; 1\; \&\; 2\; \backslash \backslash \; 3\; \&\; 4\; \backslash \backslash \; 5\; \&\; 6\; \backslash end^\; =\; \backslash begin\; 1\; \&\; 3\; \&\; 5\backslash \backslash \; 2\; \&\; 4\; \&\; 6\; \backslash end$Properties

Let and be matrices and be a scalar.Products

If is an matrix and is its transpose, then the result ofmatrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...

with these two matrices gives two square matrices: is and is . Furthermore, these products are symmetric matrices. Indeed, the matrix product has entries that are the inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

of a row of with a column of . But the columns of are the rows of , so the entry corresponds to the inner product of two rows of . If is the entry of the product, it is obtained from rows and in . The entry is also obtained from these rows, thus , and the product matrix () is symmetric. Similarly, the product is a symmetric matrix.
A quick proof of the symmetry of results from the fact that it is its own transpose:
:$\backslash left(\backslash mathbf\; \backslash mathbf^\backslash operatorname\backslash right)^\backslash operatorname\; =\; \backslash left(\backslash mathbf^\backslash operatorname\backslash right)^\backslash operatorname\; \backslash mathbf^\backslash operatorname=\; \backslash mathbf\; \backslash mathbf^\backslash operatorname\; .$
Implementation of matrix transposition on computers

On acomputer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as Computer program, pr ...

, one can often avoid explicitly transposing a matrix in memory
Memory is the faculty of the mind by which data or information is Encoding (memory), encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If Foresight (psycholo ...

by simply accessing the same data in a different order. For example, software libraries for linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, 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
A fast Fourier transform (FFT) is an algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perf ...

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
In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...

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
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, 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 denote the algebraic dual space of an - module . Let and be -modules. If is alinear map
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

, then its algebraic adjoint or dual, is the map defined by .
The resulting functional is called the pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward.
Precomposition
Precomposition with a Function (mathematics), function probab ...

of by .
The following relation characterizes the algebraic adjoint of
: for all and
where is the natural pairing (i.e. defined by ).
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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

of a topological vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

(TVS) is denoted by .
If and are TVSs then a linear map is weakly continuous if and only if , in which case we let denote the restriction of to .
The map is called the transpose of .
If the matrix describes a linear map with respect to bases of and , then the matrix 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 defines a bilinear form , with the relation . By defining the transpose of this bilinear form as the bilinear form defined by the transpose i.e. , we find that . Here, is the naturalhomomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

into the double dual.
Adjoint

If the vector spaces and have respectively nondegeneratebilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called ''Vector (mathematics), vectors'') over a Field (mathematics), field ''K'' (the elements of which are called ''scalar (mathematics), scalars''). ...

s and , a concept known as the adjoint, which is closely related to the transpose, may be defined:
If is a linear map
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

between vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s and , we define as the adjoint of if satisfies
:$B\_X\backslash big(x,\; g(y)\backslash big)\; =\; B\_Y\backslash big(u(x),\; y\backslash big)$ for all and .
These bilinear forms define an isomorphism
In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...

between and , and between and , resulting in an isomorphism between the transpose and adjoint of .
The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal
In linear algebra, two vector space, vectors in an inner product space are orthonormal if they are orthogonality, orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutual ...

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 is equal to .
In particular, this allows the orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

over a vector space with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps for which the adjoint equals the inverse.
Over a complex vector space, one often works with sesquilinear form
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

s (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
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

, the transpose of the cofactor matrix
* Conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n Complex number, complex matrix (mathematics), matrix \boldsymbol is an n \times m matrix obtained by transpose, transposing \boldsymbol and applying ...

* Moore–Penrose pseudoinverse
* Projection (linear algebra)
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces ...

References

Further reading

* * . * * * *External links

* Gilbert Strang (Spring 2010Linear Algebra

from MIT Open Courseware {{Tensors Matrices Abstract algebra Linear algebra