Matrix multiplication examples

## Permutations

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### Nimber multiplication table

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Walsh permutation; nimber multiplication; patterns

## Walsh spectrum of Boolean functions

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

The Walsh spectrum of the Boolean function  ( 1 , 0 , 1 , 0 , 0 , 1 , 1 , 0 ) {\displaystyle (1,0,1,0,0,1,1,0)}
is  ( 4 , 2 , 0 , − 2 , 0 , 2 , 0 , 2 ) {\displaystyle (4,2,0,-2,0,2,0,2)}

Compare Figure 1 in Walsh Spectrum Computations Using Cayley Graphs
by W. J. Townsend and M. A. Thornton | Walsh spectra of 8 Boolean functions
in the same small equivalence class
(including the function in the file on the left)

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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

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The 3-ary Boolean functions in ggbec O have this feature.

## LDU decomposition of a Walsh matrix

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LDU decomposition of the Walsh matrix of order 16

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.

## Product of a Walsh matrix and Gould's-Morse sequence

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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

n = 8 | n = 16

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## "n-ary Walsh matrices"

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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.

Binary Walsh matrix
white 0, red 1 | "Ternary Walsh matrix"
white 0, green 1, red 2 | "Balanced ternary Walsh matrix"
red −1, white 0, green 1

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## Quaternion group

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

Q. g. as a subgroup of SL(2,C) | Q. g. as a subgroup of SL(2,3)

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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).

Cayley table of SL(2,3)

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

Cycle graph of SL(2,3)