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How to estimate the largest eigenvalue of followed characteristics? Let $A={a_{ij}}$.

  1. Symmetric positive definite. Real. Very sparse.
  2. Diagonal elements are all positive, and off-diagonal elements are all negative.
  3. $a_{ii} $ equals to the negative of the sum of the rest elements of that row/column. $a_{ii}=-\sum^{k=n}_{k=1}a_{ik}$, $k\neq i$.
  4. $a_{ij}$ in (-1, 1].

Is this type of matrix special enough to have its own largest eigenvalue estimation?

The reason of my question is, I have 12 example matrices have the forehead mentioned characteristics, size from 30 by 30 to 3000 by 3000, but I happen to find that their largest eigenvalue only ranges in (1, 2). Therefore I am wondering if it is due to these characteristics.

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You can always apply Gershgorin's lemma: given a square matrix $A$ with elements $a_{ij}$, define radiuses

$$r_i=\sum_{j=1..n,j\ne i}|a_{ij}|.$$

Then all eigenvalues of the matrix $A$ lie in the union

$$\bigcup_{i=1..n}\{z\in \Bbb C: |z-a_{ii}|\le r_i\}.$$

In your case this results in all eigenvalues lying in the set $[0,2]$.

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