On the invertibility of symmetric Toeplitz matrices

• dwcook
In summary, the conversation discusses the conditions for the invertibility of a symmetric Toeplitz matrix, with entries from binomial coefficients, and seeking help for a proof for a family of matrices. It is known that a bisymmetric matrix has a bisymmetric inverse. The search for a simpler condition, such as a specific diagonal magnitude, is mentioned.
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]

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.

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.

1. What is a symmetric Toeplitz matrix?

A symmetric Toeplitz matrix is a square matrix where the entries are constant along both diagonals. This means that the entries are the same along the main diagonal (from top left to bottom right) and also along the secondary diagonal (from top right to bottom left).

2. How do you determine if a symmetric Toeplitz matrix is invertible?

A symmetric Toeplitz matrix is invertible if and only if its determinant is non-zero. This can be determined by using the Gershgorin circle theorem or by calculating the determinant using a formula specifically designed for symmetric Toeplitz matrices.

3. Why is the invertibility of symmetric Toeplitz matrices important?

The invertibility of symmetric Toeplitz matrices is important in various fields such as signal processing, image processing, and linear algebra. It allows for efficient computations and can also provide insights into the structure of the matrix and its properties.

4. Can a symmetric Toeplitz matrix be singular?

Yes, a symmetric Toeplitz matrix can be singular if and only if its determinant is equal to zero. This means that the matrix is not invertible and has no inverse.

5. Are there any special properties of the inverse of a symmetric Toeplitz matrix?

Yes, the inverse of a symmetric Toeplitz matrix is also a symmetric Toeplitz matrix. This means that the entries along both diagonals remain constant and the inverse matrix has the same structure as the original matrix.

• Linear and Abstract Algebra
Replies
6
Views
766
• Linear and Abstract Algebra
Replies
7
Views
817
• Linear and Abstract Algebra
Replies
11
Views
4K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
727
• Linear and Abstract Algebra
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
570
• Linear and Abstract Algebra
Replies
6
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
7
Views
2K