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- A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix

I need to prove the following:

A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds:

$$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$

and in addition the only non zero elements ##a_{i,j}## are those that ## i-1 \leq j \leq i+1##

Does anyone have ideas?

A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds:

$$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$

and in addition the only non zero elements ##a_{i,j}## are those that ## i-1 \leq j \leq i+1##

Does anyone have ideas?

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