# Questions tagged [toeplitz-matrices]

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

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### Perturbation of eigenvalues in non-Hermitian block Toeplitz matrix

Suppose we are given a non-Hermitian block Toeplitz matrix $A$ and that we perturb it by $E$, which is a matrix with only two non-zero elements in row $i$ and column $j$ and row $j$ and column $i$ ...
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### Unitary transformation to a Toeplitz matrix

Suppose it is known that a matrix $M \in \mathbb{C}^{n \times n}$ has a factorization of the form $M=UT$, where $U,T \in \mathbb{C}^{n \times n}$ and $U$ is unitary and $M$ is Toeplitz. Is it ...
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### The eigenvector of toeplitz matrix

The Toeplitz matrix is \begin{align*} T_r(a,b,c)= \begin{bmatrix} b & c \\ a & b & c\\ &a&b&c\\ &&\ddots&\ddots&\ddots\\ &&&...
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### Eigendecomposition of Hermitian Toeplitz matrices

Are their any fast methods available for full eigendecomposition of Hermitian Toeplitz matrices?
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### Pseudo-inverse of a fat Toeplitz matrix

I have a fat Toeplitz matrix, say, \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 &...
1answer
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### Do eigenvectors of a Toeplitz matrix form an orthogonal set?

It is true for a $2 \times 2$ Toeplitz matrix (put values $a$ and $b$ in the first row and $b$ and $a$ in the second and work out), but when I tried it for a $3 \times 3$, it turns out to be a bit ...
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### Represent a Toeplitz matrix in an array

I need to represent a $n \times n$ Toeplitz matrix in a $2n - 1$ array. I need to create a function that takes a pair $(i,j)$ and returns the value in the $2n - 1$ array. I am having a difficult time ...
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### Is this positive definite?

Given $n \times (n+m-1)$ Toeplitz matrices $A$ and $B$, if $AB^T$ is positive definite, how to prove that $$\left( A^T - B^T \right) \left( BB^T \right)^{-1} B + I$$ is also positive definite?
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### Inverse and multiplication of (symmetric, positive definite) Toeplitz matrices

Let $A \in \mathbb R^{n \times n}$ and $B \in \mathbb R^{n\times k}$ be two Toeplitz matrices, with $A$ symmetric and positive definite. I am searching for an elegant proof (or a counterexample) for ...
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### Are all Toeplitz matrices diagonalizable?

As in the title. Also, if anyone knows if all Hermitian-symmetric matrices with distinct diagonal elements are diagonalizable, that'd be great to know. Thanks. Edit: Never mind about the Hermitian ...
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### If $\kappa (A) > \kappa (B)$, show $\kappa (B^{-1}A) < \kappa (A)$

Let $A$ and $B$ be a toeplitz and symmetric positive definite $NxN$ matrices. If $\kappa (A) > \kappa (B)$, how to show that: $$\kappa (B^{-1}A) < \kappa (A)$$ where $\kappa$(X) is ...
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### Diagonalization of a bisymmetric matrix

Is there some way to easily diagonalize a rank-$n$ bisymmetric Toeplitz matrix with only zeros on its main diagonal? Direct calculation is out of the question. I need some trick. Addendum: I don't ...