Questions tagged [tridiagonal-matrices]
Relating to all $n\times n$ matrices $(A)$ with the property $a_{i,j}=0$ if $|j-i|>1$
174
questions
-1
votes
1
answer
37
views
How to show two matrices are similar? [closed]
Show that the matrices
A = ⎡ 1 1 1 ⎤
⎢ 1 1 1 ⎥
⎣ 1 1 1 ⎦
and B = ⎡ 3 0 0 ⎤
⎢ 0 0 0 ⎥
⎣ 0 0 0 ⎦
are similar?
I know I should show there exist P- such that A=PBP-, but how do I find P, and is there a ...
1
vote
1
answer
107
views
Maximal and minimal eigenvalues of a symmetric tridiagonal Toeplitz matrix
Given $m \times m$ symmetric tridiagonal Toeplitz matrices $$M=\begin{pmatrix}
4 & 1 & & \\ 1 & 4 & \ddots & \\ & \ddots & \ddots & 1\\ & & 1 & 4\...
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0
votes
0
answers
25
views
Trace of the square root of a tridiagonal matrix
I have the following $n \times n$ symmetric tridiagonal matrix:
\begin{equation} \label{myeq}
M = \left[ \begin{matrix}
a_{1}^{2} & -ca_{1} & & & & 0 \\
-ca_{1} & a_{2}^{2} &...
0
votes
0
answers
9
views
inverting a tridiagonal circulant matrix with alternating elements
First of all, I want to underline that my knowledge regarding matrices is extremely restricted.. It just so happens that they are popping out everywhere in a recent project of mine.
So, in this ...
1
vote
0
answers
44
views
Proving a complex block tridiagonal matrix is nonsingular
I have a complex block tridiagonal matrix that I am trying to prove has an inverse. The matrix M here:
$$M = \begin{bmatrix}
A' & B' & 0 & \cdots & 0 \\
B & A & B & \...
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0
votes
0
answers
25
views
Inversion of a Toeplitz tridiagonal matrix
Currently, I am working on solving a PDE using the finite element method (FEM) and facing the problem of finding the inverse of the following Toeplitz tridiagonal matrix
$$\mathbf{M} = \begin{pmatrix}
...
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1
vote
1
answer
31
views
Tridiagonal matrix with main diagonal equal to 1, inverse 1-norm upper bound
Description
Suppose we have the following non singular tridiagonal matrix
$$
B =
\begin{bmatrix}
1 & a_1
\\ b_2 & 1 & a_2
\\ & \ddots & \ddots & \ddots
\\ && b_{n-...
- 406
0
votes
2
answers
62
views
Recursive computation of determinant of Toeplitz tridiagonal matrix
Let a matrix be a tridiagonal matrix of size $n \times n$, with elements equal to $2$ on the main diagonal, elements equal to $1$ directly above the main diagonal, elements equal to $3$ directly below ...
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1
vote
0
answers
41
views
Using matrix to analyze the property of an ODE?
Let $a_k(t), b_k (t) \in \mathbb{R} \ (k = 1, 2, \dots, n)$ satisfy the following differential equations:
$$\begin{aligned} \dot a_k(t) &= 2 \left( b_k^2 - b_{k-1}^2 \right) \\ \dot b_k(t) &= ...
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0
votes
0
answers
17
views
Decomposition of stochastic matrices into tridiagonal ones
I was wondering if some results were known for the following problem for which I couldn't find any information, except for this similar question that unfortunately doesn't have answers.
So let's start ...
- 31
3
votes
2
answers
124
views
Toeplitz tridiagonal matrix with $0$s on main diagonal and $1$s on sub/superdiagonal has distinct eigenvalues [closed]
$$\begin{pmatrix}0&1&&&\\ 1&\ddots&\ddots&&\\ &\ddots&\ddots&\ddots&\\ &&\ddots&\ddots&1\\ &&&1&0\end{pmatrix}$$
has ...
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1
vote
2
answers
88
views
Can this $3 \times 3$ tridiagonal Toeplitz matrix be rank-$1$?
I am trying to determine whether the following tridiagonal $3 \times 3$ matrix can have a rank of $1$.
$$\begin{bmatrix}a&b&0\\b&a&b\\0&b&a\end{bmatrix}$$
For $a = b = 0$, the ...
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0
votes
0
answers
64
views
Wilkinson's matrix eigengap
Task
The $(2n+1) \times (2n+1)$ Wilkinson matrix is a symmetric tridiagonal matrix whose main diagonal is $$n, n-1, \ldots, 1, 0, 1, \ldots, n$$ and whose entries on the first diagonals above and ...
0
votes
0
answers
33
views
the spectrum of tridiagonal matrix is bounded
What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?
how could I prove that tridiagonal matrix have the smallest eigenvalue，and for any 𝜆
, the spectrum of that matrix is ...
- 1
2
votes
1
answer
90
views
How to notate off-diagonal blocks for a tridiagonal block matrix
I have a block diagonal matrix that also has partitioned identity matrices in the off-diagonal blocks, like a tridiagonal block matrix. It looks something like
$$
\begin{bmatrix}
\textbf{A} & I &...
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0
votes
0
answers
35
views
Positive definiteness of symmetric block tridiagonal matrix
Let $A\in\mathbb{R}^{n\times n}$ be the symmetric tridiagonal matrix
\begin{equation*}
A:=\begin{pmatrix}
u_1 & v_1 & \\
v_1 & u_2 & v_2 \\
& v_2 & \ddots & \ddots \\
&...
- 41
5
votes
1
answer
153
views
Analytical expression for the determinant of block tridiagonal matrix
I have a $3n\times3n$ matrix $M$ that is in the following block tridiagonal form:
$$M=\begin{pmatrix}
A & B^T & 0\\
B & A & UBU \\
0 & UB^T U & A\\
\end{pmatrix}$$
where $A,B,...
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3
votes
0
answers
60
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 &...
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0
votes
0
answers
56
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 \\...
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1
vote
0
answers
50
views
Parallel LU decomposition of diagonal matrices
I have seen efficient implementations of LU decomposition for tridiagonal and penta diagonal matrices. I have a banded diagonal matrix that is huge 100k rows by 100k columns, but each row has only 27 ...
- 11
2
votes
0
answers
87
views
What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?
$A_{nm}$ ($n,m = 0, 1, 2, \ldots$) is a symmetric, tridiagonal matrix. The diagonal elements are $A_{nn} = a_n = n + 1$, and the off-diagonal elements are $A_{n,n+1} = A_{n+1,n} = b_n = \lambda \sqrt{\...
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3
votes
1
answer
129
views
Partial derivative of a diagonal matrix w.r.t a vector
I am trying to find the second partial derivative of the function
$Y=diag\boldsymbol(S)\mathbb P diag \boldsymbol(\beta)diag^{-1}(\mathbb P^{T}\boldsymbol S +\mathbb P^{T}\boldsymbol E + \mathbb P^{T}\...
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1
vote
0
answers
81
views
Exponential of infinite dimensional matrix
Asked this on MathOverflow with some more details
I have a matrix originating from Master Equation for birth death process on semi infinite lattice. It is tridiagonal. It is not symmetric.
I wanted ...
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1
vote
1
answer
69
views
Instability of cubic splines
I'm supposed to show, that the calculation of cubic splines with tridiagonal matrices is unstable. To show this I'm supposed to consider the function $s_1: [x_0, x_n] \to \mathbb{R}$ which is a spline ...
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0
votes
1
answer
70
views
Eigenvalue Problem of a hermitian tridiagonal matrix
I want to prove that the eigenvalues of the following complex, hermitian matrices are the same
$$
A=\begin{pmatrix}
\alpha_1&\beta_1\\
\overline{\beta_1}&\alpha_2&\beta_2\\
&\overline{\...
0
votes
2
answers
227
views
Find eigenvalues and eigenvectors of particular Toeplitz matrix
Assume a matrix in this form:
$$
\begin{bmatrix}
b & c & 0 & \dots & 0 & a \\
a & b & c & 0 ...
- 81
0
votes
1
answer
34
views
Advection-Diffusion Equation with two variables
Consider the following equation:
$\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2} -u \frac{\partial T}{\partial x} -T \frac{\partial u}{\partial x}$
where $T$ is the ...
- 27
0
votes
1
answer
67
views
What is the name of the theorem of tridiagonal reduction of symmetric matrices?
I have over the last year been implementing a code to solve the many-body Schrödinger equation for applications in Nuclear physics, as part of my PhD-studies. The most important step in that code is ...
1
vote
0
answers
95
views
Proof involving tridiagonal diagonally dominant matrix
$\mathbf{Background}$
Define a finite difference grid on the rectangle $[0,X] \times [0,T]$,
$(x_j,t_n) = (x_0 + j\Delta x,n\Delta t) \text { for } 0 \leq j \leq M \text { and } 0 \leq n \leq N , \...
user715112
4
votes
1
answer
83
views
since a symmetric tridiagonal matrix contains only two distinct vectors
I don't understand meaning of "since a symmetric tridiagonal matrix contains only two distinct vectors" I write example for symmetric tridiagonal matrix. The 1st row vector and 1st column ...
user847292
1
vote
1
answer
55
views
How can one compute the determinant of a tridiagonal matrix when using integers?
My question is similar to How to compute the determinant of a tridiagonal matrix with constant diagonals?
However, all of the options seem to include computations that are not permitted for integers. ...
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1
vote
0
answers
45
views
Example of regular tridiagonal matrix $A$ with given properties
I am looking for a regular tridiagonal matrix $A$ such that at the LU-decomposition with partial column pivoting the matrices $L$ and $U$ are also tridiagonal, but with total pivoting the matrices $L$ ...
- 12.2k
0
votes
0
answers
47
views
linear transformation, is diagonalizable? all values of a?
People i have the next problem.
Let $f: \mathbb R_3 → \mathbb R_3$ be the linear transformation such that https://i.stack.imgur.com/qLTqd.png with $\mathcal B = \{(1,0, −1), (0,0,1), (0,1 , 0)\}$. The ...
1
vote
0
answers
83
views
Finding the eigenvalues of a discrete laplacian on an infinite lattice
If we define the Laplacian as a square matrix with zeroes on the diagonal, and -1 on the diagonals exactly above and below the main diagonal, and 0 everywhere else, how would one go about finding its ...
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0
votes
1
answer
50
views
How to show that two Hermitian tridiagonal matrices are similar?
Given two tridiagonal hermitian matrices A,B with $a_i\in \mathbb{R}$ and $b_i\in \mathbb{C}$ as follows
\begin{align}
A=
\begin{pmatrix}
a_{1} & |b_1| & \cdots & 0 \\
|b_1| & a_{...
- 319
3
votes
0
answers
97
views
Decomposition of a dense matrix into $m$ tridiagonal matrices
Given an $n\times n$ dense matrix $\mathbf{K}$, is there any decomposition that factorizes $\mathbf{K}$ into $m$ tridiagonal matrices such that $\mathbf{K} = \mathbf{B}_1 \, \mathbf{B}_2 \, \mathbf{B}...
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0
votes
1
answer
66
views
Diagonalize a quasi-uniform tridiagonal matrix
How to diagonalize the following matrix?
$$
A=\begin{pmatrix}
0 & a& 0 & 0 & 0 & \cdots \\
a& 0 & b & 0 & 0 & \cdots \\
0 & b & 0 & b & 0 &...
0
votes
0
answers
59
views
Eigenvalues of sum of a tridiagonal matrix and another matrix whose square is a diagonal matrix
$A$ is a tridiagonal matrix of the form $$\left( \begin{array}{ccccccccc}d_1&a&0&0&\cdots&0&0&0&0\\a&d_2&a&0&\cdots&0&0&0&0\\0&a&...
- 11
0
votes
1
answer
84
views
Show that the eigenvalues of the matrix are symmetric around origin
Let $A \in M_n (\mathbb{R})$ be a matrix whose all entries are $0$ except those on the diagonals above and below the principal diagonal, which are all $1$. That is, $a_{ij} = 1$ if $|i-j|=1$ and $0$ ...
0
votes
0
answers
47
views
How to find the eigenvalues of the following tridiagonal matrix [duplicate]
The $n$-order tridiagonal matrix is defined by $A(i,i)=0, A(i,i+1)=n-i, A(i+1,i)=i$, i.e.,
$$A=\begin{pmatrix}0&n-1\\1&0&n-2\\&2&0&n-3\\&&\ddots&\ddots&\ddots\\&...
- 246
1
vote
0
answers
95
views
Determinant of the sum of tridiagonal and anti-tridiagonal matrices
$A$ is a tridiagonal matrix and $B$ is an anti-tridiagonal matrix, both of size $n\times n$, such that $B^2$ is a diagonal matrix. Is it possible to express $\det(A+B)$ in terms of $\det(A),\det(B),n$ ...
- 11
2
votes
0
answers
54
views
similarty anti-tridiagonal?
Let $n \times n$ matrix $H_n$ be anti-tridiagonal. For example, when $n = 4$,
$$H_4 = \begin{bmatrix} 0 & 0 & a & b\\ 0 & a & b & c\\ a & b & c & 0\\ b &...
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3
votes
0
answers
90
views
Falkner Skan solution using a tridiagonal matrix method. For 0, -ve $\beta$ I get incorrect solutions?
I am trying to solve the Falkner Skan equation for wedge flows numerically, by first reducing it to a second order system and then solving a tridiagonal matrix equation iteratively till solution ...
- 131
0
votes
1
answer
74
views
similarity matrix tridiagonal
I am solving a problem, in the middle of which I reached the following block-tridiagonal matrix.
\begin{bmatrix}
0 & B_1 & &0 \\
B_1 & 0 & \ddots \\
&\ddots& \...
- 63
1
vote
0
answers
107
views
Eigenvalues of a tridiagonal matrix with $-1, 2, -1$ as entries
This problem was given as an exercise for section 6.2 of Hoffman & Kunze's book in a course in linear algebra, which I couldn't solve and I really don't know how to solve it, may be because I don'...
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8
votes
2
answers
358
views
Calculate determinant of a continuant matrix with variable elements?
One of the problems in Muir's Treatise on determinants in exercise set XXX asks to evaluate the determinant of a continuant matrix
$$
f_n(x,y)=\left|\begin{array}{cccc}
x & 1 & 0 & 0 & ...
- 293
0
votes
0
answers
88
views
Tridiagonal matrices and vector space dimension
"The set of all $n$ × $n$ tridiagonal matrices forms a 3n-2 dimensional vector space."(Wikipedia)
I'm familiar with what a tridiagonal matrix is but I'm not sure why this specifically is ...
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0
votes
1
answer
216
views
What is a simple definition of Lanczos iteration that is understandable?
I have seen that Lanczos's Algorithm can be used to tri-diagonalize a matrix but all of the definitions I have seen of it have been very complicated to understand.
- 25
1
vote
1
answer
60
views
Special tridiagonal matrix name and property
I was wondering if there is a special name for the following kind of tridiagonal matrices ? And if yes, are there any books or articles which talk about their properties ?
\begin{pmatrix}
\alpha_1 &...
- 25
3
votes
1
answer
97
views
Elementary block matrix operations to block-triangularize block tridiagonal matrix
For some flow modeling purpose, the system $(S)$ defined as $MX=L$ is solved, where $M$ is a tridiagonal block matrix defined as
$$ M= \begin{pmatrix}B_1& C_1 &0 &0 \\ A_2 & B_2 & ...
- 33