Hadamard-Matrix

A Hadamard matrix is a matrix of the degree that includes only the numbers, and a coefficient, and in addition, all columns are mutually orthogonal, as are all row.

Hadamard matrices are named after the French mathematician Jacques Hadamard.

Properties

From the orthogonality of the lines and columns following a Hadamard matrix, the relation:

The transposed matrix referred to and is the identity matrix. This equation can also be used for the definition of the Hadamard matrices, as in all matrices whose entries consist only of numbers and only Hadamard matrices comply with this equation.

It can be shown that Hadamard matrices only, or can exist with.

Include the first column and the first row of only entries, it means normalizes the matrix.

Construction

There are several methods to construct Hadamard matrices. Two of them are described below:

Design by Sylvester

This construction goes back to the English mathematician James J. Sylvester. If a Hadamard matrix of degree, so can thus follows a Hadamard matrix construct the degree:

The orthogonality is easy to check:

Walsh matrices

This results, for example, named after the mathematician Joseph L. Walsh sequence of matrices ( Walsh matrices):

The Walsh - Hadamard matrices are normalized on the degree matrices, each row is a Walsh function.

Structure via the Legendre symbol

We first defined in this construction the Jacobsthal matrix of degree (where is an odd prime ) with the help of the Legendre symbol:

Is now with so applies

And

Wherein the one designated matrix, in which all entries are 1. Now we construct the Hadamard matrix of degree:

Also here one can verify that this is a Hadamard matrix ( and use ):

So constructed matrices called Hadamard matrices of Paley - type, after the English mathematician Raymond Paley.

The Hadamard conjecture

It is believed (but could not be proved) that to each number there is at least one Hadamard matrix. This assumption is probably due to Paley. With the two methods mentioned above can be generated for all numbers Hadamard matrices of the mold or for one prime number. There are other methods, however, can not be thus cover all the possibilities. So no Hadamard matrix was found to to 2005. 1977 was the question yet for unexplained.

Applications

• Hadamard transform, a discrete transform of the region of the Fourier analysis is used Hadamard matrices.
• In statistics, they are used to calculate variances of variables.
• In discrete mathematics, certain block plans that Hadamard block diagrams, constructed from Hadamard matrices.