# Questions tagged [tridiagonal-matrices]

Relating to all $n\times n$ matrices $(A)$ with the property $a_{i,j}=0$ if $|j-i|>1$

210 questions
Filter by
Sorted by
Tagged with
58 views

### Give algorithm for finding bidiagonal matrix similar to triangular matrix

Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and outputs a bidiagonal matrix $B$ which $T$ is similar to? We may assume that $T$ is invertible. ...
• 8,175
40 views

### Leading eigenvalue of a strange tridiagonal matrix with matrix sub-blocks

Let's assume the following crazy matrix P = \begin{pmatrix} \mathbb{0}_{1\times 1} & \alpha \mathbb{1}_{1\times N} & \mathbb{0}_{1\times\frac{N(N-7)}{2!}} & \...
• 31
26 views

### Linear phase shifts in the exponential of special tridiagonal matrices

I've been working on a physics problem that has led me to make the following numerical observation. Let $L$ be an $N\times N$ real-symmetric tridiagonal matrix whose diagonal entries are zero. This ...
• 215
47 views

### Is the product $B D B^T$ always a symmetric tridiagonal matrix? Where $D$ is a diagonal matrix and $B$ a sparse matrix.

I have a diagonal matrix ${\bf D}_{n \times n}$ and a rectangular matrix ${\bf B}_{m \times n}$ where $n \gg m$. All but $m$ rows of ${\bf B}$ have non-zero elements. These $m$ rows have only six non-...
• 1,241
48 views

### Eigenvalues of tridiagonal Toeplitz matrix with diagonals $1$, $0$, and $-1$

Consider a matrix $A \in M_n(\mathbb{R})$ with entries denoted by $A=[a_{ij}]$. When $i=j+1$, $a_{ij}=1$, and when $i=j-1$, $a_{ij}=-1$, with all other entries being zero. Determine the eigenvalues of ...
982 views

• 305
96 views

### Eigenvalues of a symmetric tridiagonal matrix

I'm looking for the eigenvalues of the following symmetric tridiagonal matrix \begin{pmatrix} a & z & 0 & 0 & 0 \\ z & b & z & 0 & 0 \\ 0 & z & 0 & z & ...
107 views

### Calculating the characteristic polynomial of a block tridiagonal Toeplitz Symmetric matrix

I am trying to calculate the characteristic polynomial of a block tridiagonal matrix and I need some help. This matrix is a representation of a tight-binding Hamiltonian of a finite grid of graphene, ...
112 views

• 155
1 vote
33 views

### Eigenvalue bound for a entrywise bounded trigonal positive definite matrix

Given I have a matrix $A\in\mathbb{R}^{n\times n}$ and it is tridiagonal and positive definite such that $b_{ij}$ must be zero if $|i-j|>1$. Furthermore, $0<C_1<a_{ij}<C_2$ for $|i-j|\le 1$...
• 381
1 vote
132 views

1 vote
191 views

• 13.9k
135 views

### Are tridiagonal stochastic matrices irreducible?

According to Wikipedia, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal and the supradiagonal. To my understanding, in a tridiagonal ...
• 55
1 vote
72 views

• 111
193 views

• 71
126 views

### How to find the eigenvalues of a block tridiagonal Toeplitz matrix?

I have a block tridiagonal Toeplitz matrix $$M=\begin{bmatrix} A & Z & O\\ Y & A & Z\\ O & Y & A\end{bmatrix}$$ where A=\begin{bmatrix} 0 & 1 & 0 & 1\\ 1 & 0 &...
78 views

### Eigenvalues of a certain symmetric tridiagonal Toeplitz matrix

Is there any way that can explicitly calculate eigenvalues (or at least the largest eigenvalue) of the following $n \times n$ symmetric matrix: \begin{pmatrix} 1 & 1 & 0 & 0 & \cdots \\...
• 55
1 vote