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An  matrix
 is a Hadamard matrix of order  if the entries of  are either  or  and such that
 where  is the transpose of  and  is the order  identity matrix.
In other words, an  matrix with only  and  as its elements is Hadamard if the inner product of two distinct rows is 0 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
Property 1:
The order of a Hadamard matrix is  or  where  is an integer.
Property 2:
If the rows and columns of a Hadamard matrix are permuted, the matrix remains Hadamard.
Property 3:
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.
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