Example
For example, consider the 2×4 matrix:
In row-major format, this would be stored in computer memory as the sequence (10, 11, 12, 13, 14, 15, 16, 17), i.e. the two rows stored consecutively. If we transpose this, we obtain the 4×2 matrix:
which is stored in computer memory as the sequence (10, 14, 11, 15, 12, 16, 13, 17).
| Position | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Original storage | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| Transposed storage | 10 | 14 | 11 | 15 | 12 | 16 | 13 | 17 |
If we number the storage locations 0 to 7, from left to right, then this permutation consists of four cycles:
- (0), (1 2 4), (3 6 5), (7)
That is, the value in position 0 goes to position 0 (a cycle of length 1, no data motion). And the value in position 1 (in the original storage: 10, 11, 12, 13, 14, 15, 16, 17) goes to position 2 (in the transposed storage 10, 14, 11, 15, 12, 16, 13, 17), while the value in position 2 (10, 11, 12, 13, 14, 15, 16, 17) goes to position 4 (10, 14, 11, 15, 12, 16, 13, 17), while position 4 (10, 11, 12, 13, 14, 15, 16, 17) goes back to position 1 (10, 14, 11, 15, 12, 16, 13, 17). Similarly for the values in position 7 and positions (3 6 5).
Read more about this topic: In-place Matrix Transposition
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