Solution of a certain NxN matrix, when N->∞

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In summary, the conversation discusses a math problem involving a NxN square matrix and finding the solution vector for N->∞. The matrix is a Vandermonde matrix and can be explicitly inverted, but that may not be the best approach. The task can be rephrased to finding a polynomial that satisfies certain conditions. The sampling points for this polynomial are all on a specific curve.
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onkel_tuca
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Hello fellow nerds,

I've come across a math problem, where I'd like to find the solution vector of a NxN square matrix. It is possible to find a solution for a given N, albeit numbers in the matrix become very large for any N>>1, and numbers in the solution vector become very small. So it's not easy to compute solutions. Anyhow I realized that the components of the solution vector seem to converge and I'm asking myself if it's possible to write down the solution for N->∞.

Here's the matrix:

\begin{equation}
\begin{pmatrix}
1&-2&[-2]^2&\dots&[-2]^{N-1}\\
1&-6&[-6]^2&\dots&[-6]^{N-1}\\
1&-12&[-12]^2&\dots&[-12]^{N-1}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
1&-N(N+1)&[-N(N+1)]^2&\dots&[-N(N+1)]^{N-1}
\end{pmatrix}
\begin{pmatrix}
a_1\\
a_3\\
a_5\\
\vdots\\
a_{2N-1}
\end{pmatrix}=
\begin{pmatrix}
1/3\\
1/5\\
1/7\\
\vdots\\
\frac 1 {2N+1}
\end{pmatrix}
\end{equation}

or:

\begin{eqnarray}
\sum_{k=1}^{N}a_{2k-1} \left[-l(l+1)\right]^{k-1}=\frac 1{2l+1}\;,
\end{eqnarray}

for all l≤N.

I'm looking for the vector a (the numbering of the index is indeed arbitrary). For instance for N=1, one finds a1=1/3. For N=2 one finds a1=2/5 and a3=1/30. And so on. I already found that a1→1/2 for N→∞. When I look at the numerical solutions, the other ai also seem to converge slowly, see the following plot:

agshku78.gif


But I cannot think of a way to find these values without resorting to numerics.

Cheers, Max
 
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  • #2
onkel_tuca said:
But I cannot think of a way to find these values without resorting to numerics.
Your matrix is a so-called Vandermonde matrix. A lot is known about matrices of this type, so now you have a term to search for. Since in the second column the entries are all distinct, it can be explicitly inverted, see the references in the link. (Inverting the matrix may not be the best idea, however.)
 
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  • #3
Thanks for the hint! This means I can rephrase the task to:

Find the polynomial
$$P(x)=a_1+a_3 x+a_5 x^2+...\;,$$
where
$$P(-l(l+1))=\frac 1 {2l+1}$$
for all natural l>0.
The sampling points are all on the curve
$$f(x)=\frac 1 {\sqrt{1-4x}}\;.$$
 
Last edited:

FAQ: Solution of a certain NxN matrix, when N->∞

1. What is the solution of a certain NxN matrix when N approaches infinity?

The solution of a matrix when the size of the matrix (N) approaches infinity is known as the limit of the matrix. This means that the values in the matrix will continue to approach a certain value as N gets larger and larger.

2. Can the solution of an NxN matrix be calculated when N is infinitely large?

No, the solution of a matrix cannot be calculated when N is infinitely large. This is because the values in the matrix will continue to approach a certain value but will never reach it, making it impossible to find a definitive solution.

3. How does the solution of an NxN matrix change as N gets larger?

As N gets larger, the solution of the matrix will approach a certain value but will never reach it. This value is known as the limit of the matrix and can be calculated using various mathematical techniques.

4. Is there a specific method to find the solution of an NxN matrix when N->∞?

Yes, there are various mathematical techniques that can be used to find the limit of a matrix when N approaches infinity. These include taking the determinant of the matrix, using eigenvalues and eigenvectors, and using matrix transformations.

5. Can the solution of a certain NxN matrix be used to solve real-world problems?

Yes, the solution of a matrix can be applied to various real-world problems such as optimization, physics, and engineering. The limit of a matrix can help us understand how a system behaves as it approaches infinity, which can be useful in many different applications.

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