Matrix multiplication examples
Nimber multiplication table
Walsh spectrum of Boolean functions
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.
LDU decomposition of a Walsh matrix
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
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-ary Walsh matrices"
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:
The elements of F3 are represented by:
The background color tells the order of an element: