On the invertibility of symmetric Toeplitz matrices

[dwcook]

I am curious if anyone knows conditions on the invertibility of a symmetric Toeplitz matrix. In my research, I have a symmetric Toeplitz matrix with entries coming from the binomial coefficients.

Any help would be appreciated.

Ex:
[6 4 1 0 0]
[4 6 4 1 0]
[1 4 6 4 1]
[0 1 4 6 4]
[0 0 1 4 6]

 

[marcusl]

In general, a matrix must be nonsingular to possesses an inverse. This means that the rows or columns must be linearly independent. A nonsingular matrix cannot have a zero eigenvalue.

It is known that a bisymmetric (symmetric about each diagonal) matrix such as yours has a bisymmetric inverse, should the inverse exist.

 

[dwcook]

I have gone through a standard graduate level linear algebra text and have checked all of the usual methods (finding the eigenvalues, determinant, et c.). This works fine for individual cases, but I am trying to prove this for a family of square matrices.

In particular, I was hoping someone knew of a paper in which someone showed the symmetric Toeplitz matrices are invertible if they satisfy some simpler condition, e.g., the main diagonal has a certain magnitude.