Questions tagged [hilbert-matrices]
Hilbert matrices are symmetric, positive definite and notoriously ill-conditioned matrices.
28 questions
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How to prove that this matrix is invertible by only elimination? Hoffman & Kunze exercise 1.6.12
Hoffman & Kunze exercise 1.6.12 wants a proof that this matrix is invertible
$$\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{n}
\\\\ \frac{1}{2} & \...
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Determinant of sparse Hilbert matrix
It is known that the determinant of the Hilbert matrix of dimension $N$ with elements
$$
H^N_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N
$$
namely of the form
\begin{pmatrix}
1 & \frac{1}{2} & \...
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0
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73
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Determinant of Hilbert-like matrix
It is known that the determinant of the Hilbert matrix with elements
$$
H_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N
$$
decreases exponentially to zero. It can be proven that the Hilbert-like matrix ...
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0
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1
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Prove a Hilbert-like matrix is invertible
Given an $m \times m$ matrix $M$ with $i,j$-th entry
$$
M_{i, j} = \frac{\frac{1}{i+1}+\frac{1}{j+1}}{i+j}, \quad i,j=1,\ldots,m
$$
This is a Hilbert-like matrix. I have numerically checked that it is ...
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Proving that the $n \times n$ Hilbert matrix is positive definite [duplicate]
Prove that the following matrix is positive definite. $$ A = \begin{bmatrix} 1 & \frac12 & \dots & \frac1n \\ \frac12 & \frac13 & \dots & \frac1{n+1} \\ \vdots & \vdots &...
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Solve very large Hilbert matrix programmatically
There's a problem I'm working on for school where I need to solve a very large system of equations (1million x 1million matrix) where the solution is a 1 million vector of all ones.
...
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3
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2
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How to show a Hilbert matrix is invertible?
I got the matrix for the standard inner product space on polynomial space $\mathbb{P}_n$ as
$$H_n=\begin{bmatrix}1&1/2&1/3&\cdots&1/(n+1)\\1/2&1/3&1/4&\cdots&1/(n+2)\\\...
- 947
2
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1
answer
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Hilbert matrices determinant - Recurrence relation
I have got an exercise on Hilbert matrices determinant.
Let $n \in \mathbb{N}^*$ , and $H_n$ be the Hilbert matrix of size $n \times n$.
Let's note $\Delta_{n} $ the determinant of $H_n$. I have to ...
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Prove that entries of inverse of Hilbert Matrix are all integers using results covered in a standard linear algebra course.
This is an exercise question from the first chapter of Linear Algebra by Hoffman and Kunze.
But it seems to be quite difficult to solve even with knowledge of determinants and other relevant topics.
I ...
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Determinant of special matrix [duplicate]
Consider the following matrix of harmonic series.
$$\begin{pmatrix}
1/1 & 1/2 & \ldots & 1/n \\
1/2 & 1/3 & \ldots & 1/(n+1) \\
&\ldots& \ldots &\\
1/n & 1/(n+1)...
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0
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Norm of a generalized Hilbert matrix
Given positive numbers $a_1,\ldots, a_m, b_1,\ldots, b_n$, define a $m\times n$ matrix
$X_{ij}=(a_i-b_j)/(a_i+b_j)$.
It is a bit similar to the Hilbert matrix.
The question is whether its spectral ...
- 117
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Solution to $H x = b$, where $H$ is a Hilbert Matrix
I am trying to show that the linear system $H x = b$, where $H$ is a Hilbert matrix of size $n \times n$ and
$$ b_{i} = \sum_{j=1}^{n} \frac{1}{i+j-1} $$
has the solution $x = (1,1,\dots,1)$.
...
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1
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1
answer
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What can we say about this matrix?
Let $$V = \left\{ f : [0,1] \to \mathbb R\ :\ f \text{ is a polynomial of degree} \leq n \right\}$$ Let $f_j(x)=x^j$ for $0 \le j \le n$ and let $A$ be the $(n+1) \times (n+1)$ matrix given by
$$...
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Conditional Number growth of Hilbert Matrix: Theoretical vs MatLab.
I need to investigate how the condition number of the Hilbert matrix grows with the size N. The Matlab command is: "cond(hilb(N),2)"
Compute the condition number of the Hilbert matrices Hn ∈ R, N×N, ...
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1
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How to prove that every dual linear operator of an operator on $L_2(\mathbb{R})$ shares its eigenvalues with its dual operator
I'm trying to prove that for given two dual maps $A : H \to H$ and $A^* : H^* \to H^*$ where $H = L_2(\mathbb{R})$, the set of all eigenvalues of $A$ is equal to the set of all eigenvalues of $A^*$.
...
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Growth of the condition number of Hilbert matrices
Hilbert matrices are well known to be ill-conditioned, with the columns being almost linearly dependent.
On the wikipedia page, they state that the condition number grows as $$O((1+\sqrt{2})^{4n}/\...
- 111
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vote
1
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Calculating correct slope for pitch shifting given starting resonance frequency and ending frequency
How can I calculate the correct slope for variable fac (which controls how much a signal is gradually pitch shifted), if I wanted to end at a certain frequency? An ...
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How to prove the sum of all entries of the inverse of the Hilbert matrix with order n is $n^2$?
I have read all kinds of posts relative to Hilbert matrix such as
Why does the inverse of the Hilbert matrix have integer entries?
Prove that a matrix is invertible
Why does the inverse of the ...
- 1,339
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answers
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Finding the closed form of the determinant of the Hilbert matrix
In my studies of matrix theory I came across the famous Hilbert matrix, which is a square $ n \times n $ matrix $ H $ with entries given by: $ h_{ij} = \frac{1}{i+j-1} $ and this is an example of a ...
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Solving a characteristic Polynomial of the Hilbert Matrix
I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation:
$P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$
...
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Prove that the Hilbert matrix $H_5$ has five positive eigenvalues
Prove that the $5 \times 5$ Hilbert matrix, $H_5$, has five positive eigenvalues.
I know that $\lambda$ is an eigenvalue of $H_5$ iff $$\det (\lambda I_n - H_5) = 0$$
I computed $\lambda I_n - H_5$....
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votes
1
answer
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Maximum eigenvalue and a corresponding eigenvector of an infinite Hilbert matrix
I have the following matrix
$$H=\begin{bmatrix}
1 & \frac{1}{2} & \cdots & \mbox{ad}\ +\infty\\
\frac{1}{2} & \frac{1}{3} & \cdots & \mbox{ad}\ +\infty\\
\vdots & \vdots &...
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votes
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Is the norm of the Hilbert matrix equal to $\pi$?
Let $A$ be a Hilbert matrix,
$$a_{ij}=\frac{1}{1+i+j}$$
We have the result $\| A \| \leq \pi$. I am using the subordinate norm of the Euclidean norm, i.e.,
$$\| A \| = \sup\{\langle Ax,y\rangle:\...
user146010
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Why does the inverse of the Hilbert matrix have integer entries?
Let $A$ be the $n\times n$ matrix given by
$$A_{ij}=\frac{1}{i + j - 1}$$
Show that $A$ is invertible and that the inverse has integer entries.
I was able to show that $A$ is invertible. How do I ...
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2
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The inverse of the matrix $\{1/(i+j-1)\}$ [duplicate]
Let $n$ be a positive integer. Show that the matrix
$$\begin{pmatrix}
1 & 1/2 & 1/3 & \cdots & 1/n \\
1/2 & 1/3 & 1/4 & \cdots & 1/(n+1) \\
\vdots & \vdots & \...
user62230
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Understanding a proof about Hilbert Matrix
EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO.
Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
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Some questions about Hilbert matrix
This Exercise $12$ page $27$ from Hoffman and Kunze's book Linear Algebra.
The result o Example $16$ suggests that perhaps the matrix
$$A = \begin{bmatrix} 1 & \frac{1}{2} & \ldots &...
user23505
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Prove that a matrix is invertible [duplicate]
Show that the matrix
$A = \begin{bmatrix} 1 & \frac{1}{2} & \ldots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{3} & \ldots & \frac{1}{n+1}\\ \vdots & \vdots & & \vdots \...
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