Questions tagged [hilbert-matrices]

Question 1

The continuous Hilbert matrix operator on analytic spaces of unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ is easily derived to be $H(f)(z)=\int_0^1\frac{f(t)}{1-tz}dt$. It's known that the ...

Question 2

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} & \...

Question 3

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} & \...

Question 4

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 ...

Question 5

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 ...

Question 6

Prove that the following matrix is positive definite. $$ A = \begin{bmatrix} 1 & \frac12 & \dots & \frac1n \\ \frac12 & \frac13 & \dots & \frac1{n+1} \\ \vdots & \vdots &...

Question 7

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. ...

Question 8

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)\\\...

Question 9

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 ...

Question 10

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 ...

Question 11

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)...

Question 12

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 ...

Question 13

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 $x = (1,1,\dots,1)$. ...

Question 14

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 $$...

Question 15

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 ...

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Questions tagged [hilbert-matrices]

Continuous Hilbert Matrix Operator on the upper Half-Plane

How to prove that this matrix is invertible by only elimination? Hoffman & Kunze exercise 1.6.12

Determinant of sparse Hilbert matrix

Determinant of Hilbert-like matrix

Prove a Hilbert-like matrix is invertible

Proving that the $n \times n$ Hilbert matrix is positive definite [duplicate]

Solve very large Hilbert matrix programmatically

How to show a Hilbert matrix is invertible?

Hilbert matrices determinant - Recurrence relation

Prove that entries of inverse of Hilbert Matrix are all integers using results covered in a standard linear algebra course.

Determinant of special matrix [duplicate]

Norm of a generalized Hilbert matrix

Solution to $H x = b$, where $H$ is a Hilbert Matrix

What can we say about this matrix?

Growth of the condition number of Hilbert matrices — theoretical vs Matlab

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