Questions tagged [toeplitz-matrices]

The entries of a Toeplitz matrix are constant along the diagonals parallel to the main diagonal.

196 questions
Filter by
Sorted by
Tagged with
58 views

What are the eigenvalues of a symmetric pentadiagonal Toeplitz matrix with zero tridiagonals?

I have the following symmetric pentadiagonal Toeplitz matrix, in which the superdiagonal and subdiagonal are zero. Please help me find the eigenvalues, in particular the largest one. The matrix can be ...
• 23
26 views

• 4,345
11 views

• 121
128 views

Are the columns of a random Toeplitz Matrix linearly independent?

Consider a Toeplitz matrix $T \in \mathbb{R}^{n \times p}$ with randomly independently generated entries and $n < p$. The entries of the Toeplitz matrix are generated by a continuous random ...
• 51
1 vote
154 views

Eigenvalues of two symmetric tridiagonal Toeplitz matrices

I am trying to find the eigenvalues of the following two $n \times n$ symmetric tri-diagonal Toeplitz matrices (let us call them $A$ and $B$ respectively): Note that the standard way of computing the ...
32 views

• 19
189 views

Norm of a Toeplitz operator

Hello and thank you for visiting my Stack Exchange post. I am going through the book called Introduction to large truncated Toeplitz matrices by Albrecht Böttcher & Bernd Silbermann and I am on ...
• 31
141 views

1 vote
200 views

18 views

How do i show the below proof

Let Tn and Tn' be Toeplitz matrices generated by f(θ) and f(θ + θ'). Show that for n > 0, Tn' = ΩnTnΩn where Ωn = diag(1,e−iθ',e−2iθ',...,e−i(n−1)θ').
97 views

Closed-form eigenvalues of a banded Toeplitz matrix

Let banded Toeplitz matrix $W\in \mathbb{R}^{n\times n}$ be defined by $$W_{jk} = \begin{cases} m - |j-k| & \text{ if } |j-k| \leq m \\ 0 & \text{ if } |j-k| > m \end{cases}$$ Can one get a ...
• 21
130 views

• 157
1 vote
285 views

Can this $3 \times 3$ tridiagonal Toeplitz matrix be rank-$1$?

I am trying to determine whether the following tridiagonal $3 \times 3$ matrix can have a rank of $1$. $$\begin{bmatrix}a&b&0\\b&a&b\\0&b&a\end{bmatrix}$$ For $a = b = 0$, the ...
75 views

Show last coordinate of a vector is positive

Let $T$ be a real symmetric Toeplitz matrix of dimension $n$. We write $T_i$ for the matrix with only the first $i$ rows and columns of $T$. In my implementation of the Levinson algorithm I'm building ...
• 596
1 vote
64 views

Szego's Limit Theorem for Non-Hermitian Toeplitz Matrix

Toeplitz matrices $A_{n}(f)$ is defined as: $A_{n}(f)_{i,j}=c_{i-j}$ $0\leq i,j \leq n-1$, where $c_{k}$ are Fourier Coefficients of \$f(\theta)=\sum_{k=- \infty}^{+\infty}c_{k}e^{\iota k ...