# Determinant of a specific, symmetric Toeplitz matrix

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• Rlwe
In summary, the conversation discusses the definition of a matrix ##\mathbf{B}_n## and its specific form for ##n=4##. The question is then posed about whether there is a simple closed form expression for the ratio of determinants ##\frac{\det(\mathbf{B}_n)}{\det(\mathbf{B}_{n-1})}##, to which the response is that there is not a simple expression and the determinant must be written out with all summands and factors.
Rlwe
TL;DR Summary
Calculation of a ratio of determinants for a family of specific, symmetric Toeplitz matrices
Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for example, looks like this $$\mathbf{B}_4=\begin{bmatrix} \alpha & \beta & 1 & 1 \\ \beta & \alpha & \beta & 1\\ 1 & \beta & \alpha &\beta \\ 1 & 1& \beta& \alpha \end{bmatrix}\,.$$ Is there any simple, closed form expression for a ratio of determinants ##\frac{\det(\mathbf{B}_n)}{\det(\mathbf{B}_{n-1})}##?

Not a simple one: I think you can only write out the determinant with all summands and factors and make that your formula. I doubt anything cancels out (tried a few low dimensional examples).

## 1. What is a symmetric Toeplitz matrix?

A symmetric Toeplitz matrix is a square matrix where the elements on the main diagonal are the same and the elements on either side of the diagonal are also the same. This creates a symmetry in the matrix where the upper and lower triangles are mirror images of each other.

## 2. What is the determinant of a symmetric Toeplitz matrix?

The determinant of a symmetric Toeplitz matrix can be calculated using a formula that involves the elements on the main diagonal. It is equal to the product of the elements on the main diagonal raised to the power of the matrix size minus one.

## 3. How do you find the determinant of a specific symmetric Toeplitz matrix?

To find the determinant of a specific symmetric Toeplitz matrix, you can use the formula mentioned above or you can use a computer program or calculator that has a built-in function for calculating determinants.

## 4. What is the significance of the determinant in a symmetric Toeplitz matrix?

The determinant of a symmetric Toeplitz matrix can provide information about the matrix's properties, such as whether it is invertible or singular. It is also used in various mathematical and scientific applications, such as solving systems of linear equations.

## 5. Can the determinant of a symmetric Toeplitz matrix be negative?

Yes, the determinant of a symmetric Toeplitz matrix can be negative, positive, or zero. It depends on the values of the elements on the main diagonal and the matrix size. However, if the matrix is positive definite (all eigenvalues are positive), the determinant will always be positive.

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