1. Introduction
The classical Cayley-Hamilton theorem [1-4] says that every square matrix satisfies its own characteristic equation. The Cayley-Hamilton theorem has been extended to rectangular matrices [5,6], block matrices [7,8], pairs of commuting matrices [9-11] and standard and singular two-dimensional linear systems [5,12]. The CayleyHamilton theorem has been extended to n-dimensional systems [13]. An extension of the Cayley-Hamilton theorem for 2D continuous discrete-time linear systems has been given in [14].
The Cayley-Hamilton theorem and its generalizations have been used in control systems [14,15] and also automation and control in [16,17], electronics and circuit theory [6], time-systems with delays [18-20], singular 2-D linear systems [5], 2-D continuous discrete linear systems [12], automation and electrotechnics [21], etc.
In this paper an overview of generalization of the Cayley-Hamilton theorem is presented. The linear polynomial matrix
of det
in the classical Cayley-Hamilton theorem is replaced by the general polynomial matrix
![](https://www.scirp.org/html/14-5300255\dbf07204-e261-4f08-89fc-24049cb46ee5.jpg)
where
for
are square matrices of the same order. In the Theorem 1 given below it is proved that if
and whenever for a square matrix A
implies
also. The converse of Theorem 1 is not true, is illustrated with the help of examples 1 and 2 in which the leading coefficient matrix of the polynomial matrix
may be singular or non-singular. A relation between the coefficients of the polynomial
and the coefficient matrices of
is worked out in corollaries 1, 2 and 3.
2. Preliminaries
Lemma 1. If the elements of a matrix A are polynomials in x of degree ≤ n, then A can be expressed as a polynomial matrix
in x of degree ≤ n, where the matrices
are of the same order as that of the matrix A.
Illustration 1. Let
![](https://www.scirp.org/html/14-5300255\228fe7b9-8417-4d77-8d8e-4572efb97591.jpg)
be a matrix of order 3 × 3. Then
where
;
;
and ![](https://www.scirp.org/html/14-5300255\833f0617-5396-47a9-8dbe-56b22ca74545.jpg)
Lemma 2. If A is a square matrix of order n having elements as polynomials in x each of degree ≤ m, then the elements of the adjoint of the matrix A are also polynomials in x of degree
.
Illustration 2. Let
![](https://www.scirp.org/html/14-5300255\7af2c40b-8ac3-4011-83ab-be9d4bca9c15.jpg)
be a matrix of order 3 × 3 having elements as polynomials in x of degree ≤ 4, then
where
denotes the
th element of the adjA, a polynomial in x of degree ≤ r. For instance in adjA, the element at the (2.1) th position is
.
Hence by the Lemma 1, because adjA contains elements as polynomials in x of degree ≤ 8, it implies that
, where each of the
,
is also a square matrix of order 3.
Remark 1. Prior to understand the concept in the proof of the main Theorem 1 given below, we first consider the following two illustrations of polynomial matrix
having the leading coefficient matrix singular or non-singular such that if
and for a square matrix A, whenever
![](https://www.scirp.org/html/14-5300255\b9ca50a3-6d52-44ad-9ac4-379db08f1411.jpg)
Illustration 3: Let
(2.1)
be a polynomial matrix over
for
where A2 is a non-singular matrix and
denotes the set of all 2 × 2 matrices whose elements are polynomials in x over the field F. Then there exists a matrix
such that;
![](https://www.scirp.org/html/14-5300255\a1ef2f15-79cb-4939-8967-2259c6e32744.jpg)
Also from (2.1), we have
![](https://www.scirp.org/html/14-5300255\2de21397-5ec3-4ac5-a903-970175334a65.jpg)
Hence,
implies ![](https://www.scirp.org/html/14-5300255\c76e337f-33ee-46ee-9261-b25186ec2d63.jpg)
Illustration 4: Consider the polynomial matrix
(2.2)
over
, for
;
and
, where the leading coefficient matrix A2 is singular. Then there exists a matrix
such that
![](https://www.scirp.org/html/14-5300255\78ce7ba2-7c62-4bd9-83a3-222479fe7387.jpg)
From (2.2), we have
![](https://www.scirp.org/html/14-5300255\ecbe0aaf-a518-475d-aef1-ce74633aaa0a.jpg)
As in Illustration 3, it can be easily verified that
![](https://www.scirp.org/html/14-5300255\237280bc-4f93-44ad-a696-d3e8aadf590a.jpg)
3. Main Results
Theorem 1. Let
be a polynomial matrix for
where
for
, are square matrices of order n over the field F. If
, then whenever
(Zero matrix) implies
Converse is not true.
Proof. Since
(3.1)
is itself is a matrix of order n × n having elements as polynomials in x each of degree ≤ m, therefore, using lemma 2, we have
(3.2)
Also
is a polynomial in x over
of degree ≤ mn. Therefore, using Lemma 1, we have
(3.3)
Since for any square matrix A, we have;
(3.4)
where I is the identity matrix of the same order as of A. Now using (3.4), we have
(3.5)
Therefore, using (3.1) to (3.3) above, we have from (3.5)
(3.6)
Comparing coefficients of the corresponding terms on both sides of Equation (3.6), we get
. (3.7)
Multiplying the equations in (3.7) by the matrices
![](https://www.scirp.org/html/14-5300255\4fbb51e5-c245-469d-ae87-cac4f857e742.jpg)
respectively and adding, we obtain;
![](https://www.scirp.org/html/14-5300255\de954c56-098b-48b6-bb4a-f09a2e790459.jpg)
Converse is not true. For this consider the following examples with the coefficient matrix singular and nonsingular respectively.
Example 1. Consider the function
; where
![](https://www.scirp.org/html/14-5300255\d9f97e75-1d8a-4041-9aac-10717505ac69.jpg)
Then for the scalar matrix
, we have
Whereas,
![](https://www.scirp.org/html/14-5300255\57009fd9-5e9e-43be-8d08-a787dab31341.jpg)
Example 2: Consider the function
; where
![](https://www.scirp.org/html/14-5300255\0ff43a50-9257-4c97-9d93-55d6a659666c.jpg)
Then there exist infinite number of matrices A over the complex numbers C of the form
![](https://www.scirp.org/html/14-5300255\cf67ae11-6d39-411e-92e4-63c39bade425.jpg)
or
![](https://www.scirp.org/html/14-5300255\64fbe9ab-79bb-4968-928d-6b985ce400f8.jpg)
for
, such that
but
.
For instance, if
,
, then
![](https://www.scirp.org/html/14-5300255\6a59b592-0b88-4b7e-8d4f-824ddbf59f6c.jpg)
Whereas,
![](https://www.scirp.org/html/14-5300255\9d239d8a-51f4-413b-a3ee-26c7c7c9436c.jpg)
Illustration 5. For
in Theorem 1, let
![](https://www.scirp.org/html/14-5300255\83d4813b-a900-46e3-84e5-9529a5d568ea.jpg)
be a polynomial matrix in
,where
such that
for some square matrix A of order 3.
. (3.8)
Since the elements of the matrix
are polynomials in x of degree![](https://www.scirp.org/html/14-5300255\cf5f84aa-1175-4916-a1bd-cdd3681ee367.jpg)
![](https://www.scirp.org/html/14-5300255\0b69630b-2551-4964-94fa-fc1d8eda9c27.jpg)
is a polynomial in x over the field F of degree ≤ 9. Therefore, let
(3.9)
Also each element of the
being a polynomial in x of deg ≤ 6. So by Lemma (2), let
(3.10)
Now using (3.4), we have
(3.11)
Comparing the coefficients of the equivalent powers of x on both sides, we have
(3.12)
Multiplying these equations by
respectively and adding, we get;
![](https://www.scirp.org/html/14-5300255\df8972b5-d965-466e-be1d-be076be022ae.jpg)
Corollary 1. If
and
be the polynomials given in (3.1) and (3.3) respectively, then for
.
Therefore, the constant term
of the polynomial
is the determinant of the constant term
in the polynomial matrix
.
Corollary 2. From (3.1) and (3.3), for
, we have
(3.13)
Therefore, in case for
, when
or
, then from (3.13), we have
(3.14)
Therefore, if
, then from (3.14), we get
. Hence if,
.
Thus
if the leading coefficient matrix
in
is singular.
Corollary 3. If
![](https://www.scirp.org/html/14-5300255\d2c15c9a-e0c1-40d7-925e-cba58bd70b2d.jpg)
be a bi-quadratic polynomial matrix for
![](https://www.scirp.org/html/14-5300255\7f74099c-ec48-4861-b77b-eb92e1eafaff.jpg)
and if
![](https://www.scirp.org/html/14-5300255\e9a5fe3f-76de-4c3e-a66e-772f6537300b.jpg)
Then we have,
![](https://www.scirp.org/html/14-5300255\f3fd7f87-ca6f-4054-9e8a-4a8b55a110f2.jpg)
and so on.
In general, for any
; we have pn = coefficient of
, for
;
,
.
Example 3. Consider the cubic polynomial matrix
where for
,
, if we have
![](https://www.scirp.org/html/14-5300255\0b646852-9dbe-451a-8e3d-371b04a00cb7.jpg)
where
, the coefficient of
is given by
(3.15)
It can be easily verified that
![](https://www.scirp.org/html/14-5300255\0fc04f71-de06-4a04-98ee-2c2c4c82b87e.jpg)
and
![](https://www.scirp.org/html/14-5300255\6ac797c8-04ab-4aac-944a-417dac7691bf.jpg)
Similarly coefficients of the other powers of x, i.e.,
can be found by using (3.15). For instance
![](https://www.scirp.org/html/14-5300255\1de87d16-092a-4bc5-8f53-0cc24e06cdb4.jpg)
which verifies our assertion.
4. Conclusion
The concept of the Theorem 1 given above and the relation in (3.15) can be generalized to any polynomial matrix of arbitrary degree with coefficients as square matrices of any order.
5. Acknowledgements
The author wishes to thank Dr. P. L. Sharma, Associate Professor Department of Mathematics and Statistics of the H. P. University Shimla (H.P.) India for his help and guidance. He also expresses his gratitude to the Govt. of Himachal Pradesh Department of Higher Education for granting him study leave to complete the assigned project.