# Positive Definite Matrix

## Main Question or Discussion Point

Hi,

Suppose we have:

$$q_{ij}=\int_0^1x^{i+j}\,dx$$

can we prove that

$$\mathbf{Q}=[q_{ij}]$$

is positive definite matrix? That is:

$$\mathbf{d}^T\mathbf{Q}\mathbf{d}>0$$

for all d?

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Landau
$$q_{ij}=\int_0^1x^{i+j}\,dx$$
Here i,j are natural numbers between 1 and some n? If i+j+1 is never zero, then

$$q_{ij}=\frac{1}{i+j+1}$$

or am I missing something?

The result given by Landau is called the http://en.wikipedia.org/wiki/Hilbert_matrix" [Broken]. It's a famous example of an ill-conditioned matrix. The wiki page linked to lists its properties.

As for a proof that it's positive definite. I think maybe the easiest (almost trivial) way would be to use "[URL [Broken] criterion[/URL] and induction.

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AlephZero
Homework Helper
Another easy proof is to use the fact that if every 2x2 submatrix of a matrix M is positive definite, then M is positive definite.

Here i,j are natural numbers between 1 and some n? If i+j+1 is never zero, then

$$q_{ij}=\frac{1}{i+j+1}$$

or am I missing something?
You are absolutely right. Can you go further?

Thank you Simon_Tyler and AlephZero for your replies, but I think these methods are advanced somewhat. I am taking this first course in optimization, and we use the method I mentioned in the first post.

Regards

AlephZero
Homework Helper
If you want to relate this to optimization and least squares fitting, then consider the problem of fitting a polynomial

$$P(x) = a_1 x + a_2 a^2 + \cdots + a_n x^n$$

to an arbitrary function $F(x)$ over the interval $[0,1]$. Minimize

$$\int_0^1 (P(x) - F(x))^2 dx$$

and your Hilbert matrix will appear. I can't remember much general optimization theory, but can you use this to prove the Hillbert matrix is positive definite?

If you want to relate this to optimization and least squares fitting, then consider the problem of fitting a polynomial

$$P(x) = a_1 x + a_2 a^2 + \cdots + a_n x^n$$

to an arbitrary function $F(x)$ over the interval $[0,1]$. Minimize

$$\int_0^1 (P(x) - F(x))^2 dx$$

and your Hilbert matrix will appear. I can't remember much general optimization theory, but can you use this to prove the Hillbert matrix is positive definite?
Yes I know, and from this problem exactly I got the matrix $$\mathbf{Q}$$. I did the first order necessary conditions, and moved to the second order conditions and stuck at the point at hand, which is: is Q a positive definite matrix? which means, is our solution of $$\mathbf{a}$$ is a strict relative minimum point?

Any other ideas?

Thanks

AlephZero
Homework Helper
No, we are both trying to make this too complicated.

$$\int_0^1 [P(x)]^2 dx > 0$$

for all possible values of the a's, except when all the a's are zero.

That's all there is to it.

No, we are both trying to make this too complicated.

$$\int_0^1 [P(x)]^2 dx > 0$$

for all possible values of the a's, except when all the a's are zero.

That's all there is to it.
It's pretty obvious when you put it like that!