IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 42-46 www.iosrjournals.org

On α-characteristic Equations and α-minimal Polynomial of Rectangular Matrices P. Rajkhowa1 and Md. Shahidul Islam Khan2 1 2

Department of Mathematics, Gauhati University, Guwahati-781014 Department of Mathematics, Gauhati University, Guwahati-781014

Abstract: In this paper, we study rectangular matrices which satisfy the criteria of the Cayley-Hamilton theorem for a square matrix.Various results on characteristic polynomials, characteristic equations, eigenvalues and α-minimal polynomial of rectangular matrices are proved. AMS SUBJECT CLASSIFICATION CODE: 17D20(γ,δ). Keywords: Rectangular matrix, characteristic polynomial, characteristic equation, eigenvalues and α-minimal polynomial.

I.

Definitions and introduction:

In [2], define the product of rectangular matrices A and B of order m n by A.B AB , for a fixed

rectangular matrix nm . With this product, we have A2 AA, A3 A 2 A , A 4 A3 A ,……..,

A n A n1 A . Let us consider a rectangular matrix Amn . Then we consider a fixed rectangular matrix

nm of the opposite order of A. Then A and A are both square matrices of order n and m respectively. If

m n , then A is the highest n th order singular square matrix and A is the lowest m th order square matrix forming their product. Then the matrix A I n and A I m are called the left characteristic matrix and right characteristic matrix of A respectively, where is an indeterminate. Also the determinant A I is a polynomial in of degree n, called the left characteristic polynomial of A and A I is a polynomial in of degree m, called the right characteristic polynomial of A. That is, the characteristic polynomial of singular square matrix A is called the left characteristic polynomial of A and the characteristic polynomial the square matrix A is called the right characteristic polynomial of A. The equations A I 0 and A I 0 are called the left characteristic equation and right

characteristicequation of A respectively. Then the rectangular matrix A satisfies the left characteristic equation, and the left characteristic equation of A is called the characteristic equation of A. For m n , the rectangular matrix Amn satisfies the right characteristic equation of A. So, in this case, the equation A I 0 is called the characteristic equation of A. The roots of the characteristic equation of a rectangular matrix A are called the eigenvalues of A. If is an eigenvalue of A, the matrix A I n is singular. The equation (A I n ) X 0 then

possesses a non-zero solution i.e. there exists a non-zero column vector X such that AX X . A non-zero vector X satisfying this equation is called a characteristic vector or eigenvector of A corresponding to the eigenvalue . For a rectangular matrix Amn over a field K, let J A denote the collection of all polynomial f for which f A 0 (Note that J A is nonempty, since the characteristic polynomial of A belongs to J A

).Let m be the monic polynomial of minimal degree in J A . Then m is called the minimal polynomial of A.

II.

Main results:

Theorem 2.1: If A is a rectangular matrix of order m n and is a rectangular matrix of order n m then the left characteristic polynomial of a A and right characteristic polynomial of a A are same, where A and are the transpose of A and respectively. www.iosrjournals.org

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On α-characteristic Equations and α-minimal Polynomial of Rectangular Matrices

Proof: Since A I A I

A I A I A I A I Which shows that the left characteristic polynomial of a A and right characteristic polynomial of a A are same. Similarly we can show that If A is a rectangular matrix of order m n and is a rectangular matrix of order n m then the right characteristic polynomial of a A and left characteristic polynomial of a A are same. Theorem 2.2: Let Amn , m n be a rectangular matrix. If 1 , 2 , 3 ,…., n are characteristic roots of A then the characteristic roots A Proof: Let

2

are

1 2 , 2 2 , 3 2 ,…., n 2 .

be a characteristic root of the rectangular matrix Amn , m n .Then there exists at

least a nonzero column matrix X n1 such that

AX X AAX AX AAX AX A2 X X A2 X 2 X 2 Therefore 2 is a characteristic root of A . Thus we can conclude that if 1 , 2 , 3 ,……, n

are characteristic roots of A then, 1 , 2 , 3 ,…., 2

2

2

n 2 are the characteristic roots of A 2 .

Theorem 2.3: Let A be a m n m n rectangular matrix and be a nonzero n m rectangular

matrix. If am am1 m

m1

am2 m2 ..... a1 0 is the right characteristic equation of A then

am n am1n1 am2 n2 ...... a1nm 0 is the left characteristic equation of A. Proof: Since right characteristic equation of A is the characteristic equation of the square matrix A . So A satisfies the right equation of A. That is characteristic m m1 m 2 am A am1 A am2 A ...... a1 I 0 am Am am1 Am1 am2 Am2 ...... a1 I 0 am AmAnm am1 Am1Anm am2 Am2Anm ....... a1 IAnm 0

am An am1 An1 am2 An2 ....... a1 Anm 0 (am An am1 An1 am2 An2 ....... a1 Anm ) .0

amAn am1An1 am2An2 ...... a1Anm 0 am A am1 A

n1

am2 A

n 1

Therefore am am1

am 2

n

n2

n2

...... a1 A

nm

0

...... a1

nm

0 is the characteristic equation of the singular square matrix A . Since the characteristic equation of the singular square matrix A is the left n

characteristic equation of A. Hence the result. Corollary 2.4: Let A be a m n m n rectangular matrix and be a nonzero n m rectangular

2 , 3 , ……, m are right characteristic roots of A then 1 , 2 , 3 , ……, m , are the left characteristic roots of A. That is,if A is a m n m n rectangular matrix and 0 , 0,....... 0

matrix. If 1 , n m copies

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On α-characteristic Equations and α-minimal Polynomial of Rectangular Matrices

is a nonzero n m rectangular matrix then A has at most m number of nonzero left characteristic roots of A. Corollary 2.5: Let A be a m n m n rectangular matrix and be a nonzero n m rectangular matrix. If an an1

n1

n

an2 n2 ..... a1 0 is the left characteristic equation of A then

an m an1m1 an2 m2 ...... a1mn 0 is the right characteristic equation of A. nd is a nonzero n m rectangular matrix then A satisfies its characteristic equation. Corollary 2.6: If A is a rectangular matrix of order m n and is a rectangular matrix of order n m n n 1 n2 such that n=m+1 then characteristic equation of A is an an1 an2 ...... a1 0 and the rectangular matrix A satisfies it. Theorem 2.7: If m be a minimal polynomial of a rectangular matrix A of order m n m n , then the characteristic equation of A divides m . n

Proof: Suppose m r c1r 1 c2 r 2 ...... cr 1 . Consider the following matrices:

B0 I

B1 A c1 I

B2 A c1A c2 I 2

B3 A c1 A c2A c3 I 3

2

………………………………………… ………………………………………………

Br 1 A

r 1

c1 A

r 2

Br A c1 A

r 1

r

c2 A

r 3

c2 A

r 2

..... cr 1 I

c3 A

r 3

..... cr I

Then B0 I

B1 AB0 c1 I

B2 AB1 c2 I B3 AB2 c3 I ………………………….. …………………………..

Br 1 ABr 2 cr 1 I

And

ABr 1 cr I Br

cr I [A c1 A

c2 A

[A c1A

r 2

r 1

r

r

r 1

c2A

r 2

c3A

c3 A

r 3

r 3

...... cr I ]

...... cr 1A]

( Ar c1 Ar 1 c2 Ar 2 c3 Ar 3 ...... cr 1 A)

.m A .0 0 Set B r 1 B0 r 2 B1 r 3 B2 ....... Br 2 Br 1

Then I A.B I A.(r 1 B0 r 2 B1 r 3 B2 ....... Br 2 Br 1 ) r B0 r 1 B1 AB0 r 2 B2 AB1 ....... Br 1 ABr 2 ABr 1

r I c1r 1 I c2 r 2 I ...... cr 1I

(r c1r 1 c2 r 2 ...... cr 1 ) I m I The determinant on both sides gives www.iosrjournals.org

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On α-characteristic Equations and α-minimal Polynomial of Rectangular Matrices

I A . B m I

m . n

Since B is a polynomial, I A divides m . That is the characteristic polynomial of A n

divides m . Example: Find the characteristic equation, eigenvalues, eigenvectors, minimal n

1 0 9 0 0 polynomial of the rectangular matrix A , where 0 1 . 0 4 5 0 1 Solution: The characteristic equation of A is given A I 0 , where I is the unit matrix of order three.

1 0 1 0 0 9 0 0 That is 0 1 0 4 5 0 1 0 0 0 1 0 0 1 9 0 0 o 4 5 0 0 4 5 9 0 0,9,9 Thus the eigenvalues of A are 0, 9, 9. Now, we find the eigenvectors of A corresponding to each eigenvalue in the real field. The eigenvectors of A corresponding to the eigenvalue 0 are nonzero column vector X given by the equation, A 0.I X 0 AX 0 9 0 0 x1 0 0 4 5 x 2 0 , 0 4 5 x 0 3 9 x1 0 4 x 2 5 x3 0 4 x 5x 0 3 2 9 x1 0,4 x2 5x3 0 Therefore the eigenvectors corresponding to the eigenvalue 0 are given by o 0 X 5k or 5k , where k is a real number. 4k 4 k Similarly, the eigenvectors of A corresponding to the eigenvalue 9 can be calculated. The minimal polynomial m must divide A I . Also each factor of A I that is 2

9 must also be a factor of m . Thus m f 9 or 92 . Testing f we have

and

is exactly only one of the following:

f A A2 9 A

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On α-characteristic Equations and α-minimal Polynomial of Rectangular Matrices

1 0 9 0 0 9 0 0 9 0 0 0 1 9 0 4 5 0 1 0 4 5 0 4 5 81 0 0 81 0 0 0 36 45 0 36 45 0 Thus f m 9 2 9 is the minimal of A. References:

[1]. [2].

G. L. Booth, Radicals of matrix rings, Math. Japonica, 33(1988), 325-334. H.K. Nath, A study of Gamma-Banach algebras, Ph.D. Thesis, (Gauhati University), (2001).

[3].

Jinn Miao Chen, Von Neumann regularity of two matrix

[4].

4(1987), 37-42. T. K. Dutta and H. K. Nath, On the Gamma ring of rectangular Matrices, Bulletinof Pure and Applied Science, 16E(1997), 207-216.

n,m ring M m,.n over ring, M. J. Xinjiang Univ. Nat. Sci.,

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