International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 51   | 1 | 2 |

## Section 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT)

G. Bricognea

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#### 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT)

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By virtue of definitions (i) and (ii),so that and may be represented, in the canonical bases of and , by the following matrices:

When N is symmetric, and may be identified in a natural manner, and the above matrices are symmetric.

When N is diagonal, say , then the tensor product structure of the full multidimensional Fourier transform (Section 1.3.2.4.2.4)gives rise to a tensor product structure for the DFT matrices. The tensor product of matrices is defined as follows:Let the index vectors and be ordered in the same way as the elements in a Fortran array, e.g. for with increasing fastest, next fastest, slowest; thenwhereandwhere