To solve another problem, with applications in physics and mathematics, I've constructed a sequence of seemingly simple matrices X(1),X(2), ... ,X(n), defined below. For the limit n-> infinity I want to know:

the absolutely largest (real positive) eigenvalue of X(n)

I've spend a week to find a regularity in (the process of finding the) characteristic polynomials without considerable success, so I'm feeling quite frustrated and stupid now. Does anyone BTW know an official name for this kind of matrices, with two steeper-than-usual diagonal 'strings' of ones, one below and the other above the main diagonal?

The first matrices are:

X(1)=

1 0 1

1 0 0

0 1 0

X(2)=

1 0 0 0 1 0

1 0 0 0 0 0

0 1 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 1

0 0 0 1 0 0

X(3)=

1 0 0 0 0 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 1 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 1

0 0 0 0 0 0 0 1 0 0 0 0

X(4)=

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

The examples above illustrate the regularity of the X(n):

$\displaystyle X(n):=\left( \begin{array}{cccc} A(n)& 0 & B(n) & 0 \\ 0 & I(n) & 0 & C(n) \end{array}\right)$

where

I(n) is the 2^(n-1) by 2^(n-1) unit matrix

A(n) is the 2^(n) by 2^(n-1) matrix of the form

$\displaystyle A(n)=\left( \begin{array}{ccccc} 1& & & & 0 \\ 1& & & & \\ & 1 & & & \\ & 1 & & & \\ & & \ldots & &\\& & & 1 & \\& & & 1 & \\& & & &1\\0& & & &1\\ \end{array}\right)$

B(n) is the 2^(n) by 2^(n-2) matrix of the form

$\displaystyle B(n)=\left( \begin{array}{ccccc}1& & & &0 \\0& & & & \\0& & & & \\0& & & & \\0&1 & & & \\& 0 & & & \\& 0 & & & \\& 0 & & & \\& 0 & & & \\& & \ldots & & \\& & & &1\\& & & &0\\& & & &0\\& & & &0\\0 & & & &0\\\end{array}\right)$

...and C(n) is the 2^(n-1) by 2^(n-2) matrix of the form

$\displaystyle C(n)=\left( \begin{array}{ccccc}1& & & &0 \\0& & & & \\0&1 & & & \\& 0 & & & \\& 0 & & & \\& & \ldots & & \\& & & &1\\& & & &0\\0 & & & &0\\ \end{array}\right)$

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Side remark: X(n) can also be expressed quite simply as a Khatri-Rao product.

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Side remark 2: Alternatively, finding the trace of (X(n))^q for very very large integers q and n would also be helpfull.

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Side question: what would be the best place on the internet to ask this kind of questions?

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Thanks in advance!