A Generalization of the Cayley-Hamilton Theorem

Abstract

It is proposed to generalize the concept of the famous classical Cayley-Hamilton theorem for square matrices wherein for any square matrix A, the det (A-xI) is replaced by det f(x) for arbitrary polynomial matrix f(x).

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R. Kanwar, "A Generalization of the Cayley-Hamilton Theorem," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 109-115. doi: 10.4236/apm.2013.31014.

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

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

be a matrix of order 3 × 3. Then

where

;;

and

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

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

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;

Also from (2.1), we have

Hence, implies

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

From (2.2), we have

As in Illustration 3, it can be easily verified that

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

Converse is not true. For this consider the following examples with the coefficient matrix singular and nonsingular respectively.

Example 1. Consider the function; where

Then for the scalar matrix, we have

Whereas,

Example 2: Consider the function ; where

Then there exist infinite number of matrices A over the complex numbers C of the form

or

for, such that but.

For instance, if, , then

Whereas,

Illustration 5. For in Theorem 1, let

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

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;

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

be a bi-quadratic polynomial matrix for

and if

Then we have,

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

where, the coefficient of is given by

(3.15)

It can be easily verified that

and

Similarly coefficients of the other powers of x, i.e., can be found by using (3.15). For instance

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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