The Kronecker product of two matrices
and
is defined as
In general,
.
The Hadamard Matrix is defined recursively as below:
The first column on the right of the matrix is for the
indecies of the
rows, and the second column represents the
sequency (the number of zero-crossings or sign changes) in
each row. Sequency is similar to but different from frequency in the
sense that it measures the rate of change of non-periodical signals.
The rows of the matrix can be considered as the
samples of the following waveforms:
The Hadamard matrix can also be obtained by defining its element in
the kth row and mth column of as
We can show the Hadamard matrix is orthogonal by induction.
First, it is obvious that
is orthogonal:
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In summary, the Hadamard matrix is real, symmetric, and orthogonal: