How many hadamard matrix matrices exists for size n?

Thread starter cutesteph
Start date Nov 26, 2015
Tags

Matrices Matrix

In summary, a Hadamard matrix is a square matrix with entries of either 1 or -1, where the inner product of any two distinct rows is 0 and the inner product of any row with itself is equal to the matrix size. The number of Hadamard matrices for a given size n is not known for all values of n, but a lower bound can be calculated using Sylvester's construction method. This method involves recursively constructing larger Hadamard matrices from smaller ones, starting with a 1x1 matrix and combining four copies to create larger ones. Hadamard matrices also have applications in various fields such as coding theory, cryptography, signal processing, and quantum computing, and have connections to other areas of mathematics.

Nov 26, 2015

cutesteph

Homework Statement

How many hadamard matrices exists for size n?

Homework Equations

Hadamard matrices are square matrices whose entries are either +1 or −1 and whose rows are mutually orthogonal.

The Attempt at a Solution

I am just curious how many exists for 4, 8 and in general.[/B]

Physics news on Phys.org

Nov 26, 2015

geoffrey159

0 if n is odd :-)

1. How do you define a Hadamard matrix?

A Hadamard matrix is a square matrix with entries of either 1 or -1, such that the inner product of any two distinct rows is 0 and the inner product of any row with itself is equal to the matrix size.

2. How many Hadamard matrices exist for a given size n?

The number of Hadamard matrices for a given size n is not known for all values of n. However, a lower bound for the number of Hadamard matrices can be calculated using the Sylvester's construction method, and this bound is 1 when n is a power of 2.

3. How is the Sylvester's construction method used to generate Hadamard matrices?

Sylvester's construction method involves recursively constructing larger Hadamard matrices from smaller ones. It starts with a 1x1 Hadamard matrix, and then builds larger matrices by combining four copies of the previous matrix. This method can only be used for matrix sizes that are powers of 2.

4. Are there any known Hadamard matrices for sizes that are not powers of 2?

Yes, there are Hadamard matrices for some sizes that are not powers of 2, but they are very rare and hard to find. Currently, the only known sizes for which Hadamard matrices exist are 12, 20, 28, 36, 44, 52, 60, 68, 76, 92, and 148.

5. What are the applications of Hadamard matrices?

Hadamard matrices have many applications in mathematics and engineering, including coding theory, cryptography, signal processing, and quantum computing. They are also used in experimental design and statistical analysis. Additionally, Hadamard matrices have connections to other areas of mathematics, such as number theory and combinatorics.