International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)
293
Exact Solution of a Linear-Quadratic Inverse Eigenvalue
Problem on a Certain Hamiltonian Symmetric Matrices
N. K
.Oladejo
1, S. K Amponsah
2, F.T Oduro
31
University for Development Studies, Navrongo
2,3Kwame Nkrumah University of Science and Technology, Kumasi
Abstract-- This paper investigate the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamiltonian symmetric matrices namely singular symmetric matrices of rank 1 and non-singular symmetric matrices in the neighborhood of the first type of matrices via the Newton’s iterative method.
Keywords-- Inverse, Hermiltian, eigenvalue, exact, symmetric, iterative
I. INTRODUCTION
Various theoretical results on the solvability of the inverse eigenvalue problem for Hamilton matrices together with numerical examples are systematically reviewed and discussed in respect of the inverse eigenvalue problems for certain singular and non-singular Hamilton matrices in Oduro et al (2012),Oduro (2012a, b), Baah Gyamfi (2012) as well as Oladejo et.al (2014) and (2015).This paper investigate and established the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamilton matrices consist of both singular and non-singular symmetric matrices of rank 1 via Newton’s iterative method.
Linear-quadratic optimal control system
Throughout this paper we define our notation as follows: We let
0
:
,
,
nn nm nn TQ
Q
Q
B
A
and0
:
mm TR
R
R
Where
Q
is a symmetric positive semi definite matrix andR
is a symmetric positive definite matrixWe find the linear- quadratic optimal control for the functional:
(
)
(
)
(
)
(
)
1
2
1
)
(
10
tt
T T
i
u
x
t
Qx
t
u
t
Ru
t
dt
Ix
Subject to differential equation
,
,
(
)
2
),
(
)
(
)
(
t
Ax
t
Bu
t
t
t
0t
1x
t
ix
iX
Constructing the Hamiltonian equation from equation (1) and (2) yields:
3
2
1
,
,
,
x
u
t
x
Qx
u
Ru
p
Ax
Bu
p
H
T
T
T
Given any optimal input
u
and the corresponding state
x
0
)
),
(
),
(
),
(
(
t
x
t
u
t
t
p
u
H
4
0
)
(
)
(
u
t
TR
p
t
TB
Thus:
u
(
t
)
R
1B
Tp
(
t
)
5
and the adjoint equation yields:
p
t
k
t
u
t
t
Tx
H
(
),
(
),
(
),
6
0
)
(
,
,
),
(
)
(
)
(
1
1 0
t
p
t
t
t
t
p
A
t
p
Q
t
x
T T TThus;
,
,
(
)
0
7
),
(
)
(
)
(
0 1 1
t
p
t
t
t
t
Qx
t
p
A
t
p
TConsequently;
,
, ( ) , ( ) 0
8, ) (
) ( )
( ) (
1 0
1 0
1
t p x t x t t t
t p
t x
A Q
B BR A
t p
t x dt
d
i T
T
Equation (8) is then a linear, time variant differential equation in
x
,
p
Inverse Eigenvalue-Problem for singular
2
n
2
n
symmetric matrix of rank 1
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)
294
44 43 42 41
34 33 32 31
24 23 22 21
14 13 12 11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
We assume that the singularity is due to the row dependence relations specified below:
1 1
k
R
R
i
i14 1 24 13 1 23 12 1 22 11 1
21
k
a
;
a
k
a
,
a
k
a
,
a
k
a
a
14 2 34 13 2 33 12 2 32 11 2
31
;
,
,
a
k
a
a
k
a
a
k
a
a
k
a
14 3 44 13 3 43 12 3 42 11 341
;
,
,
a
k
a
a
k
a
a
k
a
a
k
a
11 2 1 21 1 12 1
22
k
(
a
)
k
(
a
)
k
a
a
11 2 1 31 1 13 1
23
(
)
(
)
a
k
a
k
a
k
k
a
11 3 1 41 1 14 1
24
(
)
(
)
a
k
a
k
a
k
k
a
11 2 2 31 2 13 233
(
)
(
)
a
k
a
k
a
k
a
11 3 2 41 2 14 2
34
(
)
(
)
a
k
a
k
a
k
k
a
11 2 3 41 3 14 344
(
)
(
)
a
k
a
k
a
k
a
Thus:
2 3 3 2 3 1 3
3 2 2 2 2 1 2
3 1 2 1 2 1 1
3 2 1
44 43 42 41
34 33 32 31
24 23 22 21
14 13 12
11
1
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
To solve the inverse eigenvalue problem (IEP) we use the given nonzero eigenvalue as follows
)
1
(
)
(
A
a
11k
12k
22k
32tr
2 3 2 2 2 1 11
1
k
k
k
a
2 3 3 2 3 1 3
3 2 2 2 2 1 2
3 1 2 1 2 1 1
3 2 1
1
)
(
1
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
A
tr
A
Illustration
Given that
30
,
k
1
2
,
k
2
3
,
k
3
4
b
A
16
12
8
4
12
9
6
3
8
6
4
2
4
3
2
1
Hence, matrix
A
is a4
4
singular symmetric matrix which has been reconstructed from the given nonzero eigenvalue and the prescribed dependence relation parameters.Inverse Eigenvalue Problem for a non-singular
4
4
symmetric matrix- Newton’s method
We construct a characteristic (Polynomial) function of the diagonal elements of the matrix
44 43 42
41
34 33
32 31
24 23
22 21
14 13
12 11
44 43 42 41
34 33 32 31
24 23 22 21
14 13 12 11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
In other words, consider the function with independent variables defined on 4 selected elements of matrix
A
, precisely, the diagonal elements:A
trA
a
a
a
a
f
(
11,
22,
33,
44)
2
(
)
det
Thus, given four distinct eigenvalues
1,
2,
3,
4 we have the following four (4) functions with 4 independent variables being the diagonal element ofA
33 42 21 14 32 43 21 14 42 34 21 13
44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11
44 33 22 11 1 42 21 14 32 21 13 44 21 12 33 21 12
43 34 11 44 33 22 43 33 11 44 33 11 44 21 11
33 22 11 1 2 21 12 43 34 44 33 44 22 33 22
44 11 33 11 22 11 1 3 44 33 22 11 1 4
44 33 22 11 1
) ( ) (
) (
) , , , (
a a a a a a a a a a a a
a a a a a a a a a a a a a a a a
a a a a a
a a a a a a a a a a a
a a a a a a a a a a a a a a a
a a a a
a a a a a a a a a
a a a a a a a
a a a
a a a a f
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)
295
33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 2 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 2 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 2 3 44 33 22 11 2 4 44 33 22 11 2 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f
33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 3 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 3 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 3 3 44 33 22 11 3 4 44 33 22 11 3 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f
33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 4 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 4 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 4 3 44 33 22 11 4 4 44 33 22 11 4 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f
Derivation of an Explicit Formula for the Jacobian in the 4
4 case
22 33 44
1 22 33 33 44 22 44
1 2 1 3 11 1
a
a
a
a
a
a
a
a
a
a
f
11 33 44
1 11 33 33 44 11 44
1 2 1 3 22 1
a
a
a
a
a
a
a
a
a
a
f
11 22 44
1 11 22 22 44 11 44
1 2 1 3 33 1
a
a
a
a
a
a
a
a
a
a
f
11 22 33
1
11 22 22 33 11 33
1 2 1 3 44 1
a
a
a
a
a
a
a
a
a
a
f
22 33 44
2
22 33 33 44 22 44
2 2 2 3 11 2
a
a
a
a
a
a
a
a
a
a
f
11 33 44
2
11 33 33 44 11 44
2 2 2 3 22 2
a
a
a
a
a
a
a
a
a
a
f
11 22 44
2
11 22 22 44 11 44
2 2 2 3 33 2
a
a
a
a
a
a
a
a
a
a
f
:
11 22 33
2
11 22 22 33 11 33
2 2 2 3 44 2
a
a
a
a
a
a
a
a
a
a
f
22 33 44
3
22 33 33 44 22 44
3 2 3 3 11 3
a
a
a
a
a
a
a
a
a
a
f
:
11 33 44
3
11 33 33 44 11 44
3 2 3 3 22 3
a
a
a
a
a
a
a
a
a
a
f
11 22 44
3
11 22 22 44 11 44
3 2 3 3 33 3
a
a
a
a
a
a
a
a
a
a
f
11 22 33
3
11 22 22 33 11 33
3 2 3 3 44 3
a
a
a
a
a
a
a
a
a
a
f
22 33 44
4
22 33 33 44 22 44
4 2 4 3 11 4
a
a
a
a
a
a
a
a
a
a
f
11 33 44
4
11 33 33 44 11 44
4 2 4 3 22 4
a
a
a
a
a
a
a
a
a
a
f
11 22 44
4
11 22 22 44 11 44
International Journal of Emerging Technology and Advanced Engineering
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296
44 4 33 4 22 4 11 4
44 3 33 3 22 3 11 3
44 2 33 2 22 2 11 2
44 1 33 1 22 1 11 1
a f a
f a
f a
f
a f a
f a
f a
f
a f a
f a
f a
f
a f a
f a
f a
f
J
33 11 33 22 22 11 4
33 22 11 4 2 4 3
44 11 44 22 22 11 4
44 22 11 2 2 4 3
44 11 44 33 33 11 4
44 33 11 4 2 4 3
44 22 44 33 33 22 4
44 33 22 4 2 4 3
33 11 22 11 33 22 3
33 22 11 3 2 3 3
44 22 44 11 22 11 3
44 22 11 3 2 3 3
44 11 44 33 33 11 3
44 33 11 3 2 3 3
44 22 44 33 33 22 3
44 33 22 3 2 3 3
33 11 33 22 22 11 2
33 22 11 2 2 2 3
44 11 44 22 22 11 2
44 22 11 2 2 2 3
44 11 44 33 33 11 2
44 33 11 2 2 2 3
44 22 44 33 33 22 2
44 33 22 3 2 3 3
33 11 33 22 22 11 1
33 22 11 1 2 1 3
44 11 44 22 22 11 1
44 22 11 1 2 1 3
44 11 44 33 33 11 1
44 33 11 1 2 1 3
44 11 44 33 33 11 1
44 33 22 1 2 1 3
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
a a a a a a
a a a
II. NUMERICAL EXAMPLE
Consider the nonsingular symmetric Coefficient matrix of the system:
2 22 1 2
2 1 11 1
4
2
2
x
a
x
x
x
x
a
x
Given the eigenvalues
1
1
,
2
3
i.e., given a target solution of the formt t
ve
c
ue
c
x
1
2 3Assuming the initializing matrix
4
2
2
1
4
1
) 0 (
X
Moreover,
A
(0) is singular withk
2
,
a
11
1
Also
trA
(0)
5
We first compute the values of the functions at the initial point:
11 22
1
a
,
a
f
11 22
1
11 22 122
2
1
a
a
a
a
a
6
0
)
1
(
5
1
11 22
2
a
,
a
f
11 22
2
11 22 122
2
2
a
a
a
a
a
6
0
)
3
(
5
9
6
6
)
(
X
(0)f
Using the formula for the inverse of the Jacobian matrix, we have:
2 11 2 22
1 11 1 22
11 22 2 1
1
1
a
a
a
a
a
a
J
5
1
2
2
12
1
5
1
2
2
1
11 22 2 1 1
a
a
J
Substitute the values into the Newton’s iterative equation. i.e.
)
(
)
(
( ) ( )1 ) ( ) 1
(n n n n
X
f
X
J
X
X
Then:
X
(1)
X
(0)
J
1(
X
(0))
f
(
X
(0))
6
6
5
1
2
2
12
1
4
1
) 0 (X
1 1
3 0
4 1
1 1 )
1 (
International Journal of Emerging Technology and Advanced Engineering
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297
1
2
2
1
)
(
X
1A
Hence, we obtain the (exact) solution.
III. CONCLUSION
Theoretical results on the solvability of the inverse eigenvalue problem for Hamilton matrices was systematically reviewed and discussed in respect of the inverse eigenvalue problems (IEP) for certain singular and non-singular Hermitian matrices. Base on this we have successfully investigate and established the exact solution of an inverse eigenvalue problem (IEP) on a certain Hermiltian matrices consist of both singular and non-singular symmetric matrices of rank 1 via Newton’s iterative method.
REFERENCES
[1] Bhattacharyya S P.(1991) Linear control theory; structure, robustness and optimization Journal of control system, robotics and automation. CRS Press Vol IX. San Antonio Texas.
[2] Boley D and Golub.G.H (1987).Asurvey of matrix inverse eigenvalue problem. Inverse Problems.Vol,3 pp 595–622,
[3] Cai, Y.F. Kuo, Y.C Lin W.W and Xu, S.F (2009) Solutions to a quadratic inverse eigenvalue problem, Linear Algebra and its Applications. Vol. 430. Pp.1590-1606
[4] Datta B.N and Sarkissian, D.R (2004) Theory and computations of some Inverse Eigenvalue Problems for the quadratic pencil. Journal of Contemporary Mathematics,Vol.280 221-240.
[5] Miranda.J.O.A.O. Optimal Linear Quadratic Control. Control system, Robotics and Automation.Vol.VIII INESC.JD/IST,R.Alves,Redol 9.1000-029.Lisboa, Portugal [6] Oduro F.T., A.Y. Aidoo, K.B, Gyamfi, J. and Ackora-Prah, (2012)
Solvability of the Inverse Eigenvalue Problem for Dense Symmetric Matrices‟ Advances in Pure Mathematics, Vol. 3, pp 14- 19, [7] Oladejo N.K, Amponsah S.K and Oduro F.T (2014).Linear–
Quadratics optimal Control problem(LQOCP) and the Definiteness of an inverse eigenvalue Problem (IEP) on a certain Hermitian Matrices. International Journal of Emerging Technology and Advance Engineering Vol.5,Issue 5
[8] Oladejo N.K, Amponsah S.K and Oduro F.T (2014): An inverse eigenvalue problem for linear-quadratic optimal control Problem. International Journal of mathematical Archive 5(4) 306-314. [9] Ram Y.M and El-Golhary (1996). ‘An inverse eigenvalue problem
for the symmetric tridiagonal quadratic pencil with application of damped oscillatory systems’ SIAM J. Applied Math., Vol. 56 pg. 232–244
