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Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums?

For instance, the Perron–Frobenius theorem states that, for positive matrices, the largest eigenvalue can be upper bounded by the largest row sum.

I'm hoping to find an upper bound that states something like the largest eigenvalue is upper bounded by the largest sum of absolute values of a row.

An example matrix is

 3    -2     1     0     0     0
-2     3    -2     1     0     0
 1    -2     3    -2     1     0
 0     1    -2     3    -2     1
 0     0     1    -2     3    -2
 0     0     0     1    -2     3

The above matrix has a maximum absolute row sum of 9 and a maximum eigenvalue of about 7.98.

Thanks!

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2 Answers

up vote 6 down vote accepted

Yes, it is true that the largest eigenvalue is bounded by the largest absolute row sum or column sum. You can check Gershgorin circle theorem. Actually, all the eigenvalues lie in the union of all Gershorin circles.

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Also the fact that $\rho(A) \leq |||A|||$ can be helpful here (i.e. the maximum modulus of an eigenvalue is bounded by any matrix norm).

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