Why does it say "Toeplitz matrix, so we only need to calculate the first row and column of the matrix"?

Started by fl February 3, 2019
Hi,

I know Toeplitz matrix has the same value along its diagnose line. I think the
first row value gives all the value of the Toeplitz matrix, why the author said
in his way. 


///////////////
Toeplitz matrix, so we only need to calculate the first row and column
of the matrix



Thanks for reading
fl  <rxjwg98@gmail.com> wrote:

>Toeplitz matrix, so we only need to calculate the first row and column >of the matrix
The elements of a Toeplitz matrix satisfy a recursion relationship. I'd have to look further to tell you the full details. Steve
On Sunday, February 3, 2019 at 10:21:40 PM UTC-5, fl wrote:
> Hi, > > I know Toeplitz matrix has the same value along its diagnose line. I think the > first row value gives all the value of the Toeplitz matrix, why the author said > in his way. > > > /////////////// > Toeplitz matrix, so we only need to calculate the first row and column > of the matrix > > > > Thanks for reading
Consider just a 2x2 Toeplitz matrix. So, a22 is the same as a11 (both are on the main diagonal) and you know a12 (you know the first row). But what is a21 since you say that knowledge of the first column is unnecessary?? Or are you assuming that the matrix is symmetric and so knowing the first row automatically tells you the first column as well??
On Monday, February 4, 2019 at 4:21:40 PM UTC+13, fl wrote:
> Hi, > > I know Toeplitz matrix has the same value along its diagnose line. I think the > first row value gives all the value of the Toeplitz matrix, why the author said > in his way. > > > /////////////// > Toeplitz matrix, so we only need to calculate the first row and column > of the matrix > > > > Thanks for reading
There is a close link between polynomials and Toeplitz matrices too. Especially lower triangular Toeplitz. You can interchange polynomials for Toeplitz matrices in many cases.
Am 04.02.19 um 04:21 schrieb fl:
> I know Toeplitz matrix has the same value along its diagnose line. I think the > first row value gives all the value of the Toeplitz matrix, why the author said > in his way. > > > /////////////// > Toeplitz matrix, so we only need to calculate the first row and column > of the matrix
The first row is not sufficient. Look at this visualization of a general Toeplitz matrix: https://de.wikipedia.org/wiki/Toeplitz-Matrix#/media/File:Matrix_pattern_qtl1.svg You can see that the first row tells you the upper half above the main diagonal, but not the lower half. For the lower half, you'll need the first column, too. Christian