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Chapter 6: Problem 6

Show that the notorious Hilbert matrix, with elements $$ a_{i j}=(1+i+j)^{-1} \quad(0 \leq i, j \leq n) $$ is a Gram matrix for the functions \(1, x, x^{2} \ldots, x^{n-1}\).

Short Answer

Expert verified
The Hilbert matrix is a Gram matrix for the given functions, defined by their integrals as inner products.

Step by step solution

01

Define the Inner Product

We need to define an inner product for functions. In our case, we consider the inner product \( \langle f, g \rangle = \int_0^1 f(x)g(x) \, dx \). For instance, for two functions \( f(x) = x^i \) and \( g(x) = x^j \), the inner product would be \( \langle x^i, x^j \rangle = \int_0^1 x^{i+j} \, dx \).
02

Calculate the Elements of the Gram Matrix

The elements of the Gram matrix for the functions \( 1, x, x^2, \ldots, x^{n-1} \) are given by \( \langle x^i, x^j \rangle = \int_0^1 x^{i+j} \, dx \). This can be computed as \( \frac{1}{i+j+1} \), since the integral of \( x^k \) from 0 to 1 is \( \frac{1}{k+1} \).
03

Correlate with Hilbert Matrix Elements

Notice that the computed elements \( \frac{1}{i+j+1} \) align exactly with the definition of the Hilbert matrix elements \( a_{ij} \). Specifically, for any element \( a_{ij} \), according to the Hilbert matrix definition, \( a_{ij} = \frac{1}{1+i+j} \), which matches the result obtained from our integral.
04

Conclusion

Since each element \( a_{ij} = \frac{1}{1+i+j} \) of the Hilbert matrix is the result of the inner product of the functions \( x^i \) and \( x^j \), the matrix is indeed a Gram matrix for the set of functions \( 1, x, x^2, \ldots, x^{n-1} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gram matrix
A Gram matrix is an essential concept in various mathematical fields, especially in linear algebra. It is defined based on a set of vectors and their inner products.
When we talk about the Gram matrix in the context of functions, we can imagine each function as a vector in a function space. This matrix then helps us to understand how these functions correlate or interact with one another by using their inner products.
The specific case of the Hilbert matrix illustrates this concept beautifully. When arranged with elements derived from an inner product, such as \(\), the matrix showcases the interactions of polynomial functions like \(<1, x, x^2, \ldots, x^{n-1}>\). Thus, identifying the Hilbert matrix as a Gram matrix means recognizing it as a matrix of these inner products.
Inner product
The inner product is a fundamental tool in mathematics, allowing us to measure the "angle" or "distance" between functions or vectors.
In the exercise, the inner product is defined as \(\langle f, g \rangle = \int_0^1 f(x)g(x) \, dx\). For functions like \(f(x) = x^i\) and \(g(x) = x^j\), this operation essentially integrates their product over a specific interval, providing a single scalar value.
This integral captures the "overlap" between the two functions. If the functions are more alike, the inner product is larger. This concept enables us to craft the elements of a Gram matrix, as each matrix entry reflects the inner product between pairs of functions.
Polynomial functions
Polynomial functions are expressions composed of variables raised to various powers. In mathematical analysis, they serve as the building blocks for a wide array of calculations.
Given the functions \(1, x, x^2, \ldots, x^{n-1}\), each represents a monomial term of increasing degree. These functions are crucial in forming the basis for the corresponding Gram matrix, as they showcase a clear, structured interaction determined by their powers.
When used within the exercise, such functions are evaluated through their inner products, producing the entries of the Hilbert matrix. This continuous interaction makes polynomial functions perfect for demonstrating the correlation captured in a Gram matrix.
Integral calculus
Integral calculus is the branch of mathematics concerned with accumulation, areas, and integrals. It is particularly focused on finding the total quantity where the process that leads to the quantity is given.
In the presented exercise, integral calculus specifically comes into play when calculating the inner products. The operation \(\int_0^1 x^{i+j} \, dx\) suggests integration over a bounded interval, resulting in a value \(\frac{1}{i+j+1}\).
This calculation highlights how integral calculus translates the continuous nature of function interaction into quantifiable values that build the elements of a Gram matrix, thus tying back to the structure of the Hilbert matrix.

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Most popular questions from this chapter

Find a power series for the function $$ f(x)=\int_{0}^{x} \frac{e^{t}-1}{t} d t $$ Use the series you have obtained to compute \(f(1)\) with three significant figures. (In this example, that means two decimal places.) Finally, prove that the sum of all the terms you did not include is so small that it will not affect the answer. Cultural note: The function \(f\) is an important one in applied mathematics. For further information, consult Abramowitz and Stegun [1964, chap. 5].

If \(S\) is a first-degree spline function that interpolates \(f\) at a sequence of knots \(0=t_{0}<\) \(t_{1}<\cdots

Prove that if \(f\) interpolates \(g\) at \(x_{0}, x_{1}, \ldots, x_{n}\) and if \(h\) interpolates 0 at these points, then \(f \pm c h\) interpolates \(g\) at these points.

This problem and the next deal with polynomials in \(d\) variables. Label the variables with subscripts and put them into a vector called \(x\) : $$ x=\left(x_{1}, x_{2}, \ldots, x_{d}\right) \in \mathbb{R}^{d} $$ Now we require the concept of a multi-index. This is a \(d\) -tuple of nonnegative integers: $$ \alpha=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{d}\right) \in \mathbb{Z}_{+}^{d} $$ We define $$ x^{\alpha}=x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} x_{3}^{\alpha_{3}} \cdots x_{d}^{\alpha_{d}} $$ This is called a multinomial and is the basic building block for polynomials in \(d\) variables. The order of the multi-index \(\alpha\) is defined by the equation $$ |\alpha|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{d} $$ Notice that what is called the degree of the multinomial \(x^{\alpha}\) is nothing but \(|\alpha| .\) Prove that \(x^{\alpha} x^{\beta}=x^{\alpha+\beta} .\) Prove that any polynomial in \(d\) variables having degree \(\leq k\) can be written in the form $$ \sum_{|a| \leq k} c_{\alpha} x^{a} $$

Find the interpolating polynomial for the table \begin{tabular}{c||r|r|r|r} \(x\) & 1 & 3 & 2 & 6 \\ \hline\(y\) & \(-2\) & \(-22\) & \(-1\) & \(-37\) \end{tabular}
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