## Mathematics and Computer Science

#### Event Title

The Explicit Description of the Inverse and Eigenvalues of a Specific Toeplitz Matrix Type

#### Document Type

Oral Presentation

#### Location

Indianapolis, IN

#### Subject Area

Physics, Mathematics & Computer Science

#### Start Date

10-4-2015 11:30 AM

#### End Date

10-4-2015 12:00 PM

#### Sponsor

William Johnston, Rebecca Wahl (Butler University)

#### Description

The Toeplitz Matrix, or diagonal-constant matrix, is described as a matrix in which each descending diagonal from left to right is constant. It was originally studied by Otto Toeplitz who was interested in its mathematical properties. The Toeplitz matrix has received attention recently due to the discovery that its eigenvectors can help solve other problems in related fields such as engineering. We looked at this specific type of matrix to evaluate the properties of its inverse and determinant. We used a standard n x n Toeplitz matrix of this very particular form.

Results show that we can explicitly define the entries of the inverse Tn-1. We used standard matrix multiplication in the form of Tn x T n-1 = I to prove these explicit formulas are correct. Remarkably, no matter what size n, the inverse Tn-1 has only six distinct values in its entries. Further examination of this type of Toeplitz matrix has revealed other interesting properties that relate to questions in physics. The most notable of findings is that there are all distinct eigenvectors for a toeplitz matrix of this type regardless of size n. In addition, there seems to be different properties when n is odd compared to when n is even.

The Explicit Description of the Inverse and Eigenvalues of a Specific Toeplitz Matrix Type

Indianapolis, IN

The Toeplitz Matrix, or diagonal-constant matrix, is described as a matrix in which each descending diagonal from left to right is constant. It was originally studied by Otto Toeplitz who was interested in its mathematical properties. The Toeplitz matrix has received attention recently due to the discovery that its eigenvectors can help solve other problems in related fields such as engineering. We looked at this specific type of matrix to evaluate the properties of its inverse and determinant. We used a standard n x n Toeplitz matrix of this very particular form.

Results show that we can explicitly define the entries of the inverse Tn-1. We used standard matrix multiplication in the form of Tn x T n-1 = I to prove these explicit formulas are correct. Remarkably, no matter what size n, the inverse Tn-1 has only six distinct values in its entries. Further examination of this type of Toeplitz matrix has revealed other interesting properties that relate to questions in physics. The most notable of findings is that there are all distinct eigenvectors for a toeplitz matrix of this type regardless of size n. In addition, there seems to be different properties when n is odd compared to when n is even.