Category:Single 3-bit Walsh permutations; transform; JF
137 in conjugacy class 2+4 | 531 in conjugacy class 3+3 | |
---|---|---|
binary | ||
as inverse |
There are 330 images of transformed cubes in this category: 168 for the invertible binary matrices,
and 162 for the inverses that contain negative 1s. (The 6 matrices in the symmetric subgroup do not.)
The tranformation matrix with vector as inverse is not to be confused with the inverse
of the binary transformation matrix with vector .
The Walsh permutations 137 and 136 are inverse to each other.
The inverse matrix of 137 as binary is 136 as inverse.
The inverse matrix of 136 as binary is 137 as inverse.
matrix multiplication example |
---|
Python console with NumPy (as
coordinates_neutral
array([[-1, 1, -1, 1, -1, 1, -1, 1],
[-1, -1, 1, 1, -1, -1, 1, 1],
[-1, -1, -1, -1, 1, 1, 1, 1]])
matrix_137_bin
array([[1, 1, 1],
[0, 1, 1],
[0, 0, 1]])
matrix_136_inv
array([[ 1, -1, 0],
[ 0, 1, -1],
[ 0, 0, 1]])
np.dot(matrix_137_bin, matrix_136_inv)
array([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
The matrices are applied (like functions) to the neutral coordinates. coordinates_137_bin = np.dot(matrix_137_bin, coordinates_neutral)
coordinates_137_bin
array([[-3, -1, -1, 1, -1, 1, 1, 3],
[-2, -2, 0, 0, 0, 0, 2, 2],
[-1, -1, -1, -1, 1, 1, 1, 1]])
coordinates_136_inv = np.dot(matrix_136_inv, coordinates_neutral)
coordinates_136_inv
array([[ 0, 2, -2, 0, 0, 2, -2, 0],
[ 0, 0, 2, 2, -2, -2, 0, 0],
[-1, -1, -1, -1, 1, 1, 1, 1]])
When and are inverses, then . np.array_equal(
np.dot(matrix_137_bin, coordinates_136_inv),
coordinates_neutral
)
True
np.array_equal(
np.dot(matrix_136_inv, coordinates_137_bin),
coordinates_neutral
)
True
|
Subcategories
This category has the following 2 subcategories, out of 2 total.
S
Pages in category "Single 3-bit Walsh permutations; transform; JF"
The following 2 pages are in this category, out of 2 total.
Media in category "Single 3-bit Walsh permutations; transform; JF"
The following 3 files are in this category, out of 3 total.
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JF in skeletonic cube.png 2,704 × 2,637; 123 KB
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JF in skeletonic parallelepiped.png 6,376 × 2,208; 172 KB
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Walsh permutation 534 JF as inverse; screenshot of details.png 1,626 × 807; 248 KB