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Matrix operation which flips a matrix over its diagonal
the transpose of matrices and [[Transpose of a linear map

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 'A'^(T) (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT.

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.

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.

Every linear map to the dual space u : X → X^(#) defines a bilinear form B : X × X → F, with the relation
By defining the transpose of this bilinear form as the bilinear form tB defined by the transpose ^(t)u : X^(##) → X^(#) i. e. , we find that
Here, Ψ is the natural homomorphism X → X^(##) into the double dual.