In other words, an matrix with only and as its elements is Hadamard if the inner product of two distinct rows is and the inner product of a row with itself is .
A few examples of Hadamard matrices are
These matrices were first considered as Hadamard determinants, because the determinant of a Hadamard matrix satisfies equality in Hadamard’s determinant theorem, which states that if is a matrix of order where for all and then
The order of a Hadamard matrix is or where is an integer.
If the rows and columns of a Hadamard matrix are permuted, the matrix remains Hadamard.
If any row or column is multiplied by the Hadamard property is retained.
Hence it is always possible to arrange to have the first row and first column of a Hadamard matrix contain only entries. A Hadamard matrix in this form is said to be normalized.
Hadamard matrices are common in signal processing and coding applications.
|Date of creation||2013-03-22 13:09:45|
|Last modified on||2013-03-22 13:09:45|
|Last modified by||Koro (127)|