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.

E-mail:mcintosh@servidor.unam.mx