Let
ASn[a,b] denote a set of all real nxn symmetric matrices with entries in the interval [
a,
b]. In this article, we present bounds for the second largest eigenvalue
λ
2(
A) of a real symmetric matrix
A, such that
A∈
AS3 [-b,b].
Throughout this article, denotes a set of real symmetric matrices whose entries are in the interval. Eigenvalues of any real symmetric matrix A, will be represented by
The smallest and the largest eigenvalues have been studied extensively in the recent decades. Recently, many researchers have turned attention to the second largest eigenvalue due to its applications in science and engineering. For example, thesecond largest eigenvalue governs the rate at which the statistics of the Markov chain converge to equilibrium. Here, we investigate bounds for when entries of A vary in the interval.
In 1985, Constantine [1] showed that if, then
if n is even and odd respectively. Similar results are presented in [2]. In [3], Zhan gave bounds for both the largest eigenvalue and the smallest eigenvalue when entries of A are in a general interval. In the same paper [3], Zhan posed the following problem: For a given integer j with, find
We are concerned with the case j = 2 when . We employ analytical approach discussed in [4] and the properties
to determine these bounds. The following result will prove useful later. If is any real 3 × 3 matrix such that, then [5,6]. It immediately follows that if, then
This paper is organized as follows. In Section 2, analytical method for eigenvalues of real 3 × 3 symmetric matrices is discussed. In Section 3, we derive bounds for. Finally, a numerical example is given in Section 4.
2. Analytical Calculation of Eigenvalues
A detailed description of this technique can be found in [4]. Let
be a real 3 × 3 symmetric matrix. Eigenvalues of A can be directly calculated by solving the corresponding characteristic equation
where,
Equation (2.2) is then solved by first depressing it, i.e., transforming it to the form
with,
Solutions to Equation (2.4) are given by
where,
Finally, eigenvalues of A becomes
, for (2.8)
3. Bounds for the Second Largest Eigenvalue <img src="19-7400795\63687f28-50b0-4d7a-80f0-d45a3ba7c8b0.jpg" />
Note that, corresponds to the second largest eigenvalue. We therefore determine the values of x2, p and c2 which minimizes or maximizes. However, this is not straight forward since x2, p and c2 depends on the entries of A which vary in the interval. We shall heavily rely on minimizing or maximizing.
For the lower bound we require the largest possible value of p such that x2 and are minimum. Observe that if we put, then
Setting such that
where with we obtain,
and.(3.3)
Thus, as required. These correspond to the eigenvalues:
Now, suppose there exist and such that
for some real numbers. Note that
Therefore we must have. However, this is impossible since from (1.5) we have
We thus deduce that. Equality is attained by the following matrices:
Similarly, for the upper bound, we require the largest possible value of p such that x2 and are maximum. Note that setting yields
Now, if we put such that
where with we have,
and.
Check that
60˚
and hence, corresponding to the eigenvalues:
Again, assume there exist and such that
for some real numbers. Considering the fact
we necessarily have. Again from (1.5) we obtain
This is a contradiction and hence we conclude that. Equality is attained by the following matrices:
4. Numerical Example
Let. We first consider the lower bound for. According to Equation (2.8), we require, so that
Substituting (4.1) into (2.5) yields
Now, p is maximum when. However, by noting that, we require. Thus we must have, with. Finally, from (2.6), (2.7) and (4.2), we easily have, corresponding to the eigenvalues. We now let and be eigenvalues such that for some real numbers. It immediately implies that
However, (4.3) is valid only if. Applying (1.4) results in
Note that
where is the maximum determinant of a real 3 × 3 matrix whose entries are in a unit closed disc. Thus. The minimizing matrices readily follow from (3.8). For the upper bound we set 2.3, giving
Substituting (4.4) into (2.5) results in
It is easy to check that if, such that, then. This correspond to the eigenvalues. Similarly if we let and such that
then
Check that (4.6) holds only if. However,
Thus and the maximizing matrices follow from (3.17).
5. Acknowledgements
Thanks are due to the late Professor Cecilia Mwathi for her support during the initial stages of this research.
REFERENCESReferencesG. Constantine, “Lower Bounds for the Spectra of Symmetric Matrices with Nonnegative Entries,” Linear Algebra and its Applications, Vol. 65, 1985, pp. 171-178.
doi:10.1016/0024-3795(85)90095-3R. Roth, “On the Eigenvectors Belonging to the Minimum Eigenvalue of an Essentially Nonnegative Symmetric Matrix with Bipartite Graph,” Linear Algebra and Its Applications, Vol. 118, 1989, pp. 1-10.
doi:10.1016/0024-3795(89)90569-7.X. Zhan, “Extremal Eigenvalues of Real Symmetric Matrices with Entries in an Interval,” Siam Journal of Matrix Analysis and Applications, Vol. 27, No. 3, 2006, pp. 851-860. doi:10.1137/050627812J. Kopp, “Efficient Numerical Diagonalization of 3 × 3 Hermitian Matrices,” International Journal of Modern Physics C, Vol. 19, No. 3, 2008, pp. 523-548.
doi:10.1142/S0129183108012303J. Brenner, “Hadamard Maximum Determinant Problem,” The American Mathematical Monthly, Vol. 79, No. 6, 1972, pp. 626-630.N. J. A Sloan and P. Simon, “The Encyclopaedia of Integer Sequences,” Academic Press Inc., London, 1995.
denotes a set of 
real symmetric matrices whose entries are in the interval
. Eigenvalues of any real 
symmetric matrix A, will be represented by
and the largest 
eigenvalues have been studied extensively in the recent decades. Recently, many researchers have turned attention to the second largest eigenvalue 
due to its applications in science and engineering. For example, thesecond largest eigenvalue governs the rate at which the statistics of the Markov chain converge to equilibrium. Here, we investigate bounds for 
when entries of A vary in the interval
.
, then
and the smallest eigenvalue 
when entries of A are in a general interval
. In the same paper [
, find
. We employ analytical approach discussed in [
is any real 3 × 3 matrix such that
, then 
[5,6]. It immediately follows that if
, then
. Finally, a numerical example is given in Section 4.
, for 
(2.8)
, corresponds to the second largest eigenvalue. We therefore determine the values of x2, p and c2 which minimizes or maximizes
. However, this is not straight forward since x2, p and c2 depends on the entries of A which vary in the interval
. We shall heavily rely on minimizing or maximizing
.
are minimum. Observe that if we put
, then
such that
with 
we obtain,
and
.(3.3)
, as required. These correspond to the eigenvalues:
and 
such that
. Note that
. However, this is impossible since from (1.5) we have
. Equality is attained by the following matrices:
are maximum. Note that setting 
yields
such that
with 
we have,
and
.
60˚
, corresponding to the eigenvalues:
and 
such that
. Considering the fact 
. Again from (1.5) we obtain 
. Equality is attained by the following matrices:
. We first consider the lower bound for
. According to Equation (2.8), we require
, so that
. However, by noting that
, we require
. Thus we must have
, with
. Finally, from (2.6), (2.7) and (4.2), we easily have
, corresponding to the eigenvalues
. We now let 
and 
be eigenvalues such that 
for some real numbers
. It immediately implies that
. Applying (1.4) results in
is the maximum determinant of a real 3 × 3 matrix whose entries are in a unit closed disc. Thus
. The minimizing matrices readily follow from (3.8). For the upper bound we set 
2.3, giving
, such that
, then
. This correspond to the eigenvalues
. Similarly if we let 
and 
such that
then
. However,
and the maximizing matrices follow from (3.17).