# Questions tagged [tridiagonal-matrices]

A tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.

16
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### Using permutation matrix to convert a matrix into tridiagonal matrix [closed]

Let $A \in \mathbb{R}^{n \times n}$ be a bidiagonal matrix with non zero elements on its diagonal and super diagonal. Let $B$ be defined as
$$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$....

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### Action of square root of tridiagonal matrix product on vector

Assume nonsymmetric, tridiagonal matrices $A, B \in \mathbb{R}^{n\times n}$ (where $n$ is in the order of 1000) and $A, B, AB$ are diagonalizable and have positive eigenvalues.
How do you efficiently ...

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72 views

### Relation between the algebraic multiplicity of an eigenvalue and the subdiagonal elements of a symmetric tridiagonal matrix [closed]

Show that if $T$ is a symmetric tridiagonal matrix and an eigenvalue $\lambda$ has multiplicity $k$, then at least $k−1$ subdiagonal elements of $T$ are zero.
If we consider a submatrix $B$ that has ...

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83 views

### On a recurrence relation with non-constant coefficients

I am studying a real symmetric tridiagonal matrix $J_{N+1}$ (all off-diagonal elements non-zero) of dimension $N+1$, and I would like to solve the eigenvalue problem. The point is that the ...

**1**

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**1**answer

199 views

### Upper Bounds on the Largest Eigenvalue of Jacobi Matrices

Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:
$ \begin{pmatrix}
1 & a_{1} & 0 & ... & 0 \\\
a_{1} & 1 & a_{2} & & ... \\\
0 & a_{...

**1**

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**1**answer

802 views

### Relationship between eigenvalues of summation of two matrices one is diagonal

I wonder if someone can prove/disprove the following inequality,
$\lambda_i(A+mI) \leq \lambda_i(A+K) \leq \lambda_i(A+MI)$
where $A$ is a real symmetric Metzler matrix with real and nonpositive ...

**3**

votes

**1**answer

2k views

### Eigenvalues and eigenvectors of tridiagonal matrices

What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...

**6**

votes

**1**answer

525 views

### Matrix elements of exponential of tridiagonal matrices

Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...

**7**

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**1**answer

999 views

### How can I calculate eigenvalues of a tridiagonal matrix? [closed]

Are there special methods to get exact eigenvalues of a tridiagonal matrix?

**3**

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147 views

### Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix

I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...

**9**

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**0**answers

345 views

### Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix

It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix
$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\
1 & 0 & n-2 & \dots & 0\\\
0 & ...

**11**

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**1**answer

411 views

### Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...

**4**

votes

**1**answer

326 views

### Eigenvalues of large tridiagonal matrix

Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
...

**13**

votes

**5**answers

20k views

### Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & & ... \\\
0 & b_{2} & a & ... & 0 \...

**13**

votes

**4**answers

5k views

### Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix

Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...

**6**

votes

**1**answer

706 views

### Dominant eigenvector of a real symmetric tridiagonal matrix

What is the most efficient way to calculate the dominant eigenvector of a real symmetric tridiagonal matrix? What's the corresponding time complexity bound?
Could someone give me a reference for ...