algebraic structures

[AntonioLao]

The difficulty in studying modern algebra is the need to contend with so many related algebraic structures. Understanding them is similar to solving a very complex jigsaw puzzle. Some of these are shown below

[Guille]

Actually the name given to it is now abstract algebra, because modern algebra wasn't a very good name, for if it was used in 200 years time it wouldn't be modern anymore. Although now I think it it also wouldn't be considered abstract, it would be simple algebra.

About the strcutures, isn't there a kind of catogarization of the strucuture? Were some are of group bla others of group bli and others of group ble, for example?

[AntonioLao]

I am trying to use some of these structures (shown in colors) to describe the topologies of Hadamard matrices. I am doing this without the expert helps of a mathematician and having difficulty arriving at any generalization.

[quanta07]

what about Sylvester's construction

Examples of Hadamard matrices were actually first constructed by James Joseph Sylvester. Let H be a Hadamard matrix of order n. Then the partitioned matrix


is a Hadamard matrix of order 2n. This observation can be applied repeatedly and leads to the following series of matrices.




In this manner, Sylvester constructed Hadamard matrices of order 2k for every non-negative integer k.

Sylvester's matrices have a number of special properties. They are symmetric and traceless. The elements in the first column and the first row are all positive. The elements in all the other rows and columns are evenly divided between positive and negative. Sylvester matrices are closely connected with Walsh functions.

[Guille]

Sylvester's matrices have a number of special properties. They are symmetric and traceless. The elements in the first column and the first row are all positive. The elements in all the other rows and columns are evenly divided between positive and negative. Sylvester matrices are closely connected with Walsh functions.

What about their transformations?

[quanta07]

please excuse the interuption, thought link might be helpful to Antonio and others..

What about their transformations?

Here is a link that will let you have a visual effects
Use mouse pointer to control the shape..
http://www.mathsnet.net/asa2/modules/p62transform.html

Happy Thoughts..Q7

[AntonioLao]

thought link might be helpful to Antonio and others..

Thanks for the link. Although Sylvester matrices might be useful in quantum mechanics relating to Pauli matrices and Dirac matrices, the Hadamard matrices I'm working on are symmetrical along the diagonal but not traceless (the absolute value of the trace indicates the order or dimension of each matrix). Moreover, they are square and singular (determinants are zero) but not invertible.