Hadamard Matrix/Examples/Arbitrary 2 x 2

Example of Hadamard Matrix

An arbitrary example of a $2 \times 2$ Hadamard matrix is:

$\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}$


Proof

We multiply $\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}$ by its transpose, and find:

$\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix} \begin {pmatrix} 1 & -1 \\ 1 & 1 \end {pmatrix} = \begin {pmatrix} 2 & 0 \\ 0 & 2 \end {pmatrix} = 2 \begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$

Hence the result by definition of Hadamard matrix.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hadamard matrix
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hadamard matrix
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