Background
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 (e.g. Frigo & Johnson, 2005), transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing memory locality. Since these situations normally coincide with the case of very large matrices (which exceed the cache size), performing the transposition in-place with minimal additional storage becomes desirable.
Also, as a purely mathematical problem, in-place transposition involves a number of interesting number theory puzzles that have been worked out over the course of several decades.
Read more about this topic: In-place Matrix Transposition
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