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Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there exist $j$ and $k$, $j\neq k$ such that $a_j=a_k\leq 1$. $a_j$ and $a_k$ are the smallest diagonal entries.

I'm interested under what supplemental conditions can such a matrix have the smallest eigenvalue equal to 0?

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could you give some motivation/background? – Vladimir Dotsenko Feb 25 '12 at 13:26
The matrix $T$ is the linearization of a tridiagonal cooperative dynamical system around an equilibrium point. – Andreea Feb 25 '12 at 14:07
up vote 8 down vote accepted

To simplify things a little, I describe conditions under which the smallest eigenvalue is strictly positive. These can be adjusted to get equality to zero.

Necessary and sufficient conditions for positive definiteness of the tridiagonal matrix in question are described below.

Definition (Chain Sequence). A sequence $\lbrace x_k \rbrace_{k > 0}$ is a chain sequence if there exists another sequence $\lbrace y_k \rbrace_{k\ge 0}$ such that \begin{equation*} x_k = y_k(1-y_{k-1}), \end{equation*} where $y_0 \in [0,1)$ and $y_k \in (0,1)$ for $k > 0$.

By the Wall-Wetzel Theorem, your tridiagonal matrix is positive definite if and only if

\begin{equation*} \left\lbrace \frac{1}{a_ka_{k+1}} \right\rbrace_{k=1}^{n-1} \end{equation*}

is a chain sequence.

Example. In particular, if the entries of the matrix satisfy,

\begin{equation*} 0 < \frac{1}{a_ka_{k+1}} < \frac{1}{4\cos^2\left(\frac{\pi}{n+1}\right)},\quad k=1,\ldots,n-1, \end{equation*} then it is positive definite.

For additional information and details about this material, please see:

  1. M. Andelic, and C. M. Da Fonesca. Sufficient conditions for positive definiteness of tridiagonal matrices revisited. (2010).
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Thank you very much for pointing out this approach and the paper. – Andreea Feb 25 '12 at 19:02
You are welcome; also note that the Example gives a sufficient (not necessary) condition, while the theorem cited is an IFF. – Suvrit Feb 25 '12 at 19:40

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