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Whereas it is clear that the definition of r in the last section yields the largest eigenvalue with a positive eigenvector, we would like to know that it is the largest possible eigenvalue, and that it is unique. This information can be obtained somewhat indirectly by noticing that any eigenvector lacking zero components defines an invertible diagonal matrix R, containing those same components on its diagonal. Let X be the eigenvector, U the vector with unit elements. Then X = RU, so that the eigenvalue equation
is equivalent to
which means that is the common row sum of the matrix . Since the properties of boundary eigenvalues of Gerschgorin disks establish the desired result for the equivalent matrix they establish it for M as well.
Since M does not necessarily have uniform row sums, we are now interested in relating to the row sums, the largest of which we know to be an upper bound to , yet unequal to it. Here the variational definition of r is useful; let us write to acknowledge the dependence of r on M, and increase M in any way whatsoever, even by increasing one single matrix element. Then each quantity must strictly increase, and with it both the minima and maxima, leading to a strictly increased maximum eigenvalue.
At this point we could alter some of the matrix elements of M to obtain a new matrix A, whose row sums were uniformly the minimal row sum of M. It is important to do this exclusively by reducing matrix elements in M. Conversely, we could selectively increase the elements of M to obtain a matrix B whose row sums were uniformly the row sums of M, but we already have the result which this would imply.
Applying the theorem on boundary eigenvalues to the matrices A, M, and B, we obtain both upper and lower bounds to , defined as r above:
However both inequalities are strict unless the bounds coincide. Although an estimate of the gaps would be useful, often the assurance that they exist is sufficient.