Matrix multiplication examples
Permutations[edit  edit source]
Nimber multiplication table[edit  edit source]
Walsh permutation; nimber multiplication; patterns
Walsh spectrum of Boolean functions[edit  edit source]
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 3ary Boolean functions in ggbec O have this feature. 
LDU decomposition of a Walsh matrix[edit  edit source]
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'sMorse sequence.
Product of a Walsh matrix and Gould'sMorse sequence[edit  edit source]
Concider a Walsh matrix of order 2^{n}
and a column vector with the first 2^{n} values from Gould's sequence
with the signs distributed like the ones in Thue–Morse sequence sequence.
Their product always has the first 2^{n} values from A048883 (like Gould's sequence, but with powers of 3 instead of 2)
and the signs are distributed like:
 the zeros in ThueMorse sequence for odd n
 the ones in ThueMorse sequence for even n
"nary Walsh matrices"[edit  edit source]
The product of matrices made of consecutive numbers in the nbased numeral system gives an "nary 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.
Quaternion group[edit  edit source]
The quaternion group can be defined via matrix multiplication in different ways:
SL(2,3) 

The 2^{2} matrices with entries from F_{3} and determinant 1 form the special linear group SL(2,3). The elements of F_{3} are represented by:
The background color tells the order of an element:
