Closed-form determinant of a hermitian banded toeplitz matrix

In summary, the participants are discussing math problems and one person shares a problem involving a complex-valued matrix with three non-zero constant diagonals. The given matrix is a hermitian matrix with the superdiagonal elements being conjugates of the subdiagonal elements. The question is about finding the determinant or eigenvalues of the matrix in closed form. Some hints and possible approaches are also provided. The conversation ends with a summary of a calculator program that can be used to expand the determinant in terms of the first row.
  • #1
phd_student
1
0
Hello everyone,

I found that you're actively discussing math problems here and thought to share my problem with you.

[Givens:]
I have a specially structured complex-valued [itex]n \times n[/itex] matrix, that has only three non-zero constant diagonals (the main diagonal, the [itex]j^{th}[/itex] subdiagonal and the [itex]j^{th}[/itex] superdiagonal), [itex]1 \leq j \leq n-1[/itex]. Moreover, it is a hermitian matrix, where the element composing the superdiagonal is actually the conjugate of that of the subdiagonal. For example, if [itex]n=7, j=3[/itex], the matrix is given by:

\begin{eqnarray}
A &=& \left[\begin{array}{ccccccc}
a &0 &0 &b^* &0 &0 &0 \\
0 &a &0 &0 &b^* &0 &0 \\
0 &0 &a &0 &0 &b^* &0 \\
b &0 &0 &a &0 &0 &b^* \\
0 &b &0 &0 &a &0 &0 \\
0 &0 &b &0 &0 &a &0 \\
0 &0 &0 &b &0 &0 &a
\end{array} \right]
\end{eqnarray}.

[Question:] I want to get the determinant, or the eigenvalues in closed form.

[Some hints:]
- It is clear that the determinant will be only a function of [itex]a[/itex], [itex]b[/itex], the shift [itex]j[/itex] and the order of the matrix, [itex]n[/itex].
- The matrix has the following properties:
1- It is a sparse Toeplitz matrix, that has only three non-zero diagonals.
2- It is a hermitian matrix.
3- It can be regarded as a special banded matrix, with zero diagonals inside the band.
4- We can also consider it as a diagonally dominant matrix. However, neglecting [itex]b[/itex] may not give a good approximation.
- A Tridiagonal Toeplitz matrix (for the special case when [itex]j=1[/itex]) already has a known closed form expression for its eigen values, and consequently the determinant which is their direct product. It would be helpful also if we can express this shift in the diagonals as a certain simple operator, and use the known results of the tridiagonal case.

Any ideas?
 
Last edited:
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  • #2
Unfortunately there are more than three non-zero (general) diagonals. Nevertheless, the matrix is so sparse, that we can develop the determinant after, e.g. the first row and just calculate it.

A calculator program
https://matrixcalc.org/en/det.html#...,c,0,0,a,0},{0,0,0,c,0,0,a}}expand-along-row1got
$$
a^7−2a(b^*b)^3+5a^3(b^*b)^2−4a^5(b^*b)
$$
which gives the characteristic polynomial with ##a \longmapsto a-x##.
 

1. What is a closed-form determinant?

A closed-form determinant is a mathematical expression that allows for the direct calculation of the determinant of a matrix, without having to perform any row operations or use a determinant expansion formula.

2. What is a hermitian matrix?

A hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the elements of the matrix are symmetrical across the main diagonal and the complex conjugate of each element is equal to the corresponding element.

3. What does it mean for a matrix to be banded?

A banded matrix is a matrix where the non-zero elements are confined to a diagonal band around the main diagonal. The width of this band is known as the bandwidth. Banded matrices are typically used to represent sparse matrices, which have a large number of zero elements.

4. What is a Toeplitz matrix?

A Toeplitz matrix is a special type of banded matrix where the elements along each diagonal are constant. In other words, the elements are shifted by a constant value as you move along the diagonals. Toeplitz matrices are often used in signal processing and time series analysis.

5. How is the determinant of a hermitian banded Toeplitz matrix calculated?

The determinant of a hermitian banded Toeplitz matrix can be calculated using a closed-form expression known as the Cauchy-Binet formula. This formula involves taking the product of the determinants of smaller matrices, making it a more efficient method for calculating the determinant compared to other methods such as Gaussian elimination.

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