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Wed, 25 Feb 2009 01:35:04 +0000
Computation Hermite Canonical Form
http://www.mathworks.com/matlabcentral/newsreader/view_thread/245279#630608
Titus Adelani
Can anyone help me with the procedure for computing Hermite Canonical Form of a matrix that is not square? <br>
Here is the definition of a matrix that is in Hermite Canonical Form as given by Rao C<br>
1. the diagonal elements are zeros(0s) or ones (1s)<br>
2. the matrix is upper triangula such that<br>
(a) if a row has a zero on the diagonal, then all the other elements in that row are zeros<br>
(b) if a column has one on the diagonal, then all the other elements in that column are zeros<br>
Can anyone tell me if Hermite Canonical Form the same as Hermite Normal Form?<br>
.

Wed, 25 Feb 2009 03:36:02 +0000
Re: Computation Hermite Canonical Form
http://www.mathworks.com/matlabcentral/newsreader/view_thread/245279#630617
Roger Stafford
"Titus Adelani" <umadelat@cc.umanitoba.ca> wrote in message <go2787$6tl$1@fred.mathworks.com>...<br>
> Can anyone help me with the procedure for computing Hermite Canonical Form of a matrix that is not square? <br>
> Here is the definition of a matrix that is in Hermite Canonical Form as given by Rao C<br>
> 1. the diagonal elements are zeros(0s) or ones (1s)<br>
> 2. the matrix is upper triangula such that<br>
> (a) if a row has a zero on the diagonal, then all the other elements in that row are zeros<br>
> (b) if a column has one on the diagonal, then all the other elements in that column are zeros<br>
> Can anyone tell me if Hermite Canonical Form the same as Hermite Normal Form?<br>
<br>
It isn't clear what you are asking, Titus. How can we answer your question of how to compute a Hermite canonical matrix when there are infinitely many possible even of a given size? You can randomly choose 1's and 0's along the diagonal, and, by the criteria you have described, that forces some but not all elements above the diagonal to be 0's. The other upper elements can be entirely arbitrary. Below the diagonal they must of course all be 0.<br>
<br>
As to whether these matrices can be nonsquare, I am not familiar with them being other than square. There is a theorem that any Hermite canonical matrix must be idempotent, but clearly that couldn't apply to nonsquare matrices because squaring them would be meaningless.<br>
<br>
My understanding is that the Hermite normal form is different from this. See the site:<br>
<br>
<a href="http://mathworld.wolfram.com/HermiteNormalForm.html">http://mathworld.wolfram.com/HermiteNormalForm.html</a><br>
<br>
Roger Stafford

Wed, 25 Feb 2009 06:56:01 +0000
Re: Computation Hermite Canonical Form
http://www.mathworks.com/matlabcentral/newsreader/view_thread/245279#630636
Titus Adelani
Thanks Roger,<br>
Here is what I want to do. I have nonsquare matrix that can be viewed as concatenation of two square matrices. I need to reduce this matrix to its Hermite canonical form (as defined earlier). The procedure is somehow similar to reducing a matrix to its row echelon form using rref command in MATLAB. The idea is to obtain a kind of pseudoinverse (Rao inverse) of a nonsquare matrix.<br>
What I need to know is the procedure/command(s) in MATLAB to accomplish this reduction (that is reduction of a matrix to its hermite canonical form).<br>
<br>
Thank you.<br>
"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <go2eb2$q2j$1@fred.mathworks.com>...<br>
> "Titus Adelani" <umadelat@cc.umanitoba.ca> wrote in message <go2787$6tl$1@fred.mathworks.com>...<br>
> > Can anyone help me with the procedure for computing Hermite Canonical Form of a matrix that is not square? <br>
> > Here is the definition of a matrix that is in Hermite Canonical Form as given by Rao C<br>
> > 1. the diagonal elements are zeros(0s) or ones (1s)<br>
> > 2. the matrix is upper triangula such that<br>
> > (a) if a row has a zero on the diagonal, then all the other elements in that row are zeros<br>
> > (b) if a column has one on the diagonal, then all the other elements in that column are zeros<br>
> > Can anyone tell me if Hermite Canonical Form the same as Hermite Normal Form?<br>
> <br>
> It isn't clear what you are asking, Titus. How can we answer your question of how to compute a Hermite canonical matrix when there are infinitely many possible even of a given size? You can randomly choose 1's and 0's along the diagonal, and, by the criteria you have described, that forces some but not all elements above the diagonal to be 0's. The other upper elements can be entirely arbitrary. Below the diagonal they must of course all be 0.<br>
> <br>
> As to whether these matrices can be nonsquare, I am not familiar with them being other than square. There is a theorem that any Hermite canonical matrix must be idempotent, but clearly that couldn't apply to nonsquare matrices because squaring them would be meaningless.<br>
> <br>
> My understanding is that the Hermite normal form is different from this. See the site:<br>
> <br>
> <a href="http://mathworld.wolfram.com/HermiteNormalForm.html">http://mathworld.wolfram.com/HermiteNormalForm.html</a><br>
> <br>
> Roger Stafford