Matrix multiplication examples

Walsh permutation; nimber multiplication; patterns

The Walsh spectrum of a Boolean function is the product of it's binary string representation and a Walsh matrix.

The background pattern of white and red squares in the resulting matrix shows the binary Walsh spectra. In the following cases, they form binary Walsh matrices:

sec matrix     *     binary Walsh matrix       =        binary Walsh matrix
The 3-ary Boolean functions in ggbec O have this feature.

Positive numbers are green, the zero white, negatives red.
The ones in the lower and upper triangular matrices form Sierpinski triangles.
The entries of the diagonal matrix are from Gould's-Morse sequence.

Concider a Walsh matrix of order 2ⁿ
and a column vector with the first 2ⁿ values from Gould's sequence
with the signs distributed like the ones in Thue–Morse sequence sequence.

Their product always has the first 2ⁿ values from  A048883 (like Gould's sequence, but with powers of 3 instead of 2)
and the signs are distributed like:

• the zeros in Thue-Morse sequence for odd n
• the ones in Thue-Morse sequence for even n

The product of matrices made of consecutive numbers in the n-based numeral system gives an "n-ary Walsh matrix" , when modulo n operations are used. In the following files the result for normal operations is shown in light gray numbers.

In each row and column, except the one with only zeros, there is an equal number of entries for the same value.

The quaternion group can be defined via matrix multiplication in different ways:

SL(2,3)

The 22 matrices with entries from F3 and determinant 1 form the special linear group SL(2,3). It has 24 elements, and one of its subgroups is the quaternion group (fields with dark background). The elements of F3 are represented by: empty circle: 0 green dot: 1 red dot: 2          (as 2 = -1 in F3, these are (-1,0,1)-matrices) The background color tells the order of an element: dark gray: 1 light gray: 2 blue: 4 yellow: 3 white: 6