Largest eigenvalue

Gerschgorin estimates are somewhat easier to work with for positive matrices. Either of the two forms may be used

1. all eigenvalues lie within disks centered at the origin, whose radii are either the different row sums, or the different column sums. Whichever of the two yields the more restrictive results may be chosen.
2. all the eigenvalues lie within disks centered at the diagonal matrix elements, whose radii are the sums of the remaining elements in the respective row or column. Again, the more restrictive result may be used.

For positive matrices the additional information is available that the maximum eigenvalue must surpass the minimum row sum as well as being unable to reach the maximum row sum. Indeed the actual value is bounded away from these limits unless they coincide, there being results of various degrees of complexity for the actual bounds. The situation is described in Varga's book [18], along with references to original papers.

When working with a stochastic matrix, it is already a foregone conclusion that the maximum eigenvalue is 1.0, but the estimates are still valuable for non-stochastic matrices. The equilibrium eigenvector as well as the maximum eigenvalue can often be found by the expedient of successively squaring the matrix.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx