Hilbert matrices are symmetric, positive definite and notoriously ill-conditioned matrices.
A Hilbert matrix is a square matrix $H$ with entries
$$H_{ij} = \frac {1}{i+j-1}$$
Questions tagged [hilbert-matrices]
29 questions
34
votes
1 answer
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 show that $A^{-1}$ has integer entries?
This matrix…
user52991
- 748
20
votes
1 answer
Prove that a matrix is invertible
Show that the matrix
$A = \begin{bmatrix} 1 & \frac{1}{2} & \ldots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{3} & \ldots & \frac{1}{n+1}\\ \vdots & \vdots & & \vdots \\ \frac{1}{n} &\frac{1}{n+1} &\ldots &\frac{1}{2n-1} \end{bmatrix}$
is invertible and…
Herman Chau
- 743
8
votes
2 answers
Growth of the condition number of Hilbert matrices — 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 $H_N \in {\Bbb R}^{N \times N}$, for all $N \in \{ 1, 2,…
Di Wang
- 531
6
votes
1 answer
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
$$A = \begin{bmatrix} 1 & \frac{1}{2} & \ldots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{3} & \ldots &…
user23505
5
votes
0 answers
Proving that the $n \times n$ Hilbert matrix is positive definite
Prove that the following matrix is positive definite. $$ A = \begin{bmatrix} 1 & \frac12 & \dots & \frac1n \\ \frac12 & \frac13 & \dots & \frac1{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ \frac1n & \frac1{n+1} & \dots & \frac1{2n-1}…
Carl
- 187
5
votes
0 answers
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 on reading Hilbert's original paper on it from…
Ofir
- 6,343
4
votes
2 answers
How to show a Hilbert matrix is invertible?
I got the matrix for the standard inner product space on polynomial space $\mathbb{P}_n$ …
Madhan Kumar
- 866
3
votes
2 answers
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.,
$$\| A \| = \sup\{\langle Ax,y\rangle:\quad x,y\in\mathbb{R}^n,\quad\Vert x\Vert_2\leq…
user146010
2
votes
1 answer
Maximum eigenvalue and a corresponding eigenvector of an infinite Hilbert matrix
I have the following matrix
$$H=\begin{bmatrix}
1 & \frac{1}{2} & \cdots & \mbox{ad}\ +\infty\\
\frac{1}{2} & \frac{1}{3} & \cdots & \mbox{ad}\ +\infty\\
\vdots & \vdots & \ddots\\
\ & \ & \mbox{ad}\ +\infty
\end{bmatrix}$$
which is a Hilbert…
Samrat Mukhopadhyay
- 16,568
2
votes
1 answer
How to prove that this matrix is invertible by only elimination? Hoffman & Kunze exercise 1.6.12
Hoffman & Kunze exercise 1.6.12 wants a proof that this matrix is invertible
$$\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{n}
\\\\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots & \frac{1}{n+1}
\\\\ \vdots &…
pie
- 8,483
2
votes
1 answer
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 prove that :
$$ \Delta_{n+1} =…
Hugo Faurand
- 157
2
votes
1 answer
The inverse of the matrix $\{1/(i+j-1)\}$
Let $n$ be a positive integer. Show that the matrix
$$\begin{pmatrix}
1 & 1/2 & 1/3 & \cdots & 1/n \\
1/2 & 1/3 & 1/4 & \cdots & 1/(n+1) \\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1/n & 1/(n+1) & 1/(n+2) & \cdots & 1/(2n-1)
\end{pmatrix}$$
is…
user62230
2
votes
2 answers
Finding the closed form of the determinant of the Hilbert matrix
In my studies of matrix theory I came across the famous Hilbert matrix, which is a square $ n \times n $ matrix $ H $ with entries given by: $ h_{ij} = \frac{1}{i+j-1} $ and this is an example of a Cauchy matrix, which is a matrix $ C_n $ of the…
Croc2Alpha
- 3,949
1
vote
0 answers
Determinant of sparse Hilbert matrix
It is known that the determinant of the Hilbert matrix of dimension $N$ with elements
$$
H^N_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N
$$
namely of the form
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\
\frac{1}{2} &…
knuth
- 31
1
vote
0 answers
Determinant of Hilbert-like matrix
It is known that the determinant of the Hilbert matrix with elements
$$
H_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N
$$
decreases exponentially to zero. It can be proven that the Hilbert-like matrix with elements
$$
H_{tr} = \int_0^\infty d\omega \,…
knuth
- 31