How can one show that the following tridiagonal matrix
$$M_n= \begin{pmatrix}
-1&3&0&\dots&\dots&\dots&0\\
3&2&-1&0&&&\vdots\\
0&-1&2&-1&\ddots&&\vdots\\
\vdots&0&-1&2&\ddots&0&\vdots\\
\vdots&&\ddots&\ddots&\ddots&-1&0\\
\vdots&&&0&-1&2&-1\\
0&\dots&\dots&\dots&0&-1&2
\end{pmatrix}$$has exactly one negative eigenvalue?
The initial problem was showing it has at least some number of real positive eigenvalues (not necessarily distinct) for sufficiently large $n$, which can be solved by using Gershgorin circle theorem:
Which implies $-4\le \lambda\le6$ if I'm not mistaken, since $M_n$ has real eigenvalues (it's hermitian),
Then since the trace $\operatorname{Tr}M_n =2n-3$ of a matrix is also the sum of its eigenvalues, we must have at least $\frac{2n-3}{6}$ real positive eigenvalues, otherwise $\operatorname{Tr}M_n < 2n-3$ would be a contradiction.
But I noticed $M_n$ seems to have exactly one negative real eigenvalue, and the rest positive real eigenvalues, for every $n$.
How can we prove this strict bound?
Best Answer
THREE:
$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 3 & 1 & 0 \\ \frac{ 3 }{ 11 } & \frac{ 1 }{ 11 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} - 1 & 3 & 0 \\ 3 & 2 & - 1 \\ 0 & - 1 & 2 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 3 & \frac{ 3 }{ 11 } \\ 0 & 1 & \frac{ 1 }{ 11 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} - 1 & 0 & 0 \\ 0 & 11 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ - 3 & 1 & 0 \\ 0 & - \frac{ 1 }{ 11 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} - 1 & 0 & 0 \\ 0 & 11 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - 3 & 0 \\ 0 & 1 & - \frac{ 1 }{ 11 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} - 1 & 3 & 0 \\ 3 & 2 & - 1 \\ 0 & - 1 & 2 \\ \end{array} \right) $$
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FOUR:
$$ P^T H P = D $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ \frac{ 3 }{ 11 } & \frac{ 1 }{ 11 } & 1 & 0 \\ \frac{ 1 }{ 7 } & \frac{ 1 }{ 21 } & \frac{ 11 }{ 21 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} - 1 & 3 & 0 & 0 \\ 3 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & 3 & \frac{ 3 }{ 11 } & \frac{ 1 }{ 7 } \\ 0 & 1 & \frac{ 1 }{ 11 } & \frac{ 1 }{ 21 } \\ 0 & 0 & 1 & \frac{ 11 }{ 21 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} - 1 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ - 3 & 1 & 0 & 0 \\ 0 & - \frac{ 1 }{ 11 } & 1 & 0 \\ 0 & 0 & - \frac{ 11 }{ 21 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} - 1 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & - 3 & 0 & 0 \\ 0 & 1 & - \frac{ 1 }{ 11 } & 0 \\ 0 & 0 & 1 & - \frac{ 11 }{ 21 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} - 1 & 3 & 0 & 0 \\ 3 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \\ \end{array} \right) $$
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FIVE:
$$ P^T H P = D $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 & 0 \\ \frac{ 3 }{ 11 } & \frac{ 1 }{ 11 } & 1 & 0 & 0 \\ \frac{ 1 }{ 7 } & \frac{ 1 }{ 21 } & \frac{ 11 }{ 21 } & 1 & 0 \\ \frac{ 3 }{ 31 } & \frac{ 1 }{ 31 } & \frac{ 11 }{ 31 } & \frac{ 21 }{ 31 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} - 1 & 3 & 0 & 0 & 0 \\ 3 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 1 & 2 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & 3 & \frac{ 3 }{ 11 } & \frac{ 1 }{ 7 } & \frac{ 3 }{ 31 } \\ 0 & 1 & \frac{ 1 }{ 11 } & \frac{ 1 }{ 21 } & \frac{ 1 }{ 31 } \\ 0 & 0 & 1 & \frac{ 11 }{ 21 } & \frac{ 11 }{ 31 } \\ 0 & 0 & 0 & 1 & \frac{ 21 }{ 31 } \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} - 1 & 0 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } & 0 \\ 0 & 0 & 0 & 0 & \frac{ 41 }{ 31 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ - 3 & 1 & 0 & 0 & 0 \\ 0 & - \frac{ 1 }{ 11 } & 1 & 0 & 0 \\ 0 & 0 & - \frac{ 11 }{ 21 } & 1 & 0 \\ 0 & 0 & 0 & - \frac{ 21 }{ 31 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} - 1 & 0 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } & 0 \\ 0 & 0 & 0 & 0 & \frac{ 41 }{ 31 } \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & - 3 & 0 & 0 & 0 \\ 0 & 1 & - \frac{ 1 }{ 11 } & 0 & 0 \\ 0 & 0 & 1 & - \frac{ 11 }{ 21 } & 0 \\ 0 & 0 & 0 & 1 & - \frac{ 21 }{ 31 } \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} - 1 & 3 & 0 & 0 & 0 \\ 3 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 1 & 2 \\ \end{array} \right) $$
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SIX:
$$ P^T H P = D $$ $$\left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 & 0 & 0 \\ \frac{ 3 }{ 11 } & \frac{ 1 }{ 11 } & 1 & 0 & 0 & 0 \\ \frac{ 1 }{ 7 } & \frac{ 1 }{ 21 } & \frac{ 11 }{ 21 } & 1 & 0 & 0 \\ \frac{ 3 }{ 31 } & \frac{ 1 }{ 31 } & \frac{ 11 }{ 31 } & \frac{ 21 }{ 31 } & 1 & 0 \\ \frac{ 3 }{ 41 } & \frac{ 1 }{ 41 } & \frac{ 11 }{ 41 } & \frac{ 21 }{ 41 } & \frac{ 31 }{ 41 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} - 1 & 3 & 0 & 0 & 0 & 0 \\ 3 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & 3 & \frac{ 3 }{ 11 } & \frac{ 1 }{ 7 } & \frac{ 3 }{ 31 } & \frac{ 3 }{ 41 } \\ 0 & 1 & \frac{ 1 }{ 11 } & \frac{ 1 }{ 21 } & \frac{ 1 }{ 31 } & \frac{ 1 }{ 41 } \\ 0 & 0 & 1 & \frac{ 11 }{ 21 } & \frac{ 11 }{ 31 } & \frac{ 11 }{ 41 } \\ 0 & 0 & 0 & 1 & \frac{ 21 }{ 31 } & \frac{ 21 }{ 41 } \\ 0 & 0 & 0 & 0 & 1 & \frac{ 31 }{ 41 } \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrrr} - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{ 41 }{ 31 } & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{ 51 }{ 41 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ - 3 & 1 & 0 & 0 & 0 & 0 \\ 0 & - \frac{ 1 }{ 11 } & 1 & 0 & 0 & 0 \\ 0 & 0 & - \frac{ 11 }{ 21 } & 1 & 0 & 0 \\ 0 & 0 & 0 & - \frac{ 21 }{ 31 } & 1 & 0 \\ 0 & 0 & 0 & 0 & - \frac{ 31 }{ 41 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{ 21 }{ 11 } & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{ 31 }{ 21 } & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{ 41 }{ 31 } & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{ 51 }{ 41 } \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & - 3 & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{ 1 }{ 11 } & 0 & 0 & 0 \\ 0 & 0 & 1 & - \frac{ 11 }{ 21 } & 0 & 0 \\ 0 & 0 & 0 & 1 & - \frac{ 21 }{ 31 } & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{ 31 }{ 41 } \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrrr} - 1 & 3 & 0 & 0 & 0 & 0 \\ 3 & 2 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & - 1 & 2 \\ \end{array} \right) $$
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