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  AB , for a fixed

rectangular matrix  nm . With this product, we have A2  AA, A3  A 2 A , A 4  A3 A ,……..,

A n  A n1 A . Let us consider a rectangular matrix Amn . Then we consider a fixed rectangular matrix

 nm 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 Amn 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 Amn 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

42 | Page

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 Amn , 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 Amn , m  n .Then there exists at

least a nonzero column matrix X n1 such that

AX  X  AAX   AX     AAX   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   am1 m

m1

 am2 m2  .....  a1  0 is the right   characteristic equation of A then

am n  am1n1  am2 n2  ......  a1nm  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 m1 m 2 am  A   am1  A   am2  A   ......  a1 I  0  am Am  am1 Am1  am2 Am2  ......  a1 I  0  am AmAnm  am1 Am1Anm  am2 Am2Anm  .......  a1 IAnm  0

 am An  am1 An1  am2 An2  .......  a1 Anm  0   (am An  am1 An1  am2 An2  .......  a1 Anm )   .0

 amAn  am1An1  am2An2  ......  a1Anm  0  am A  am1 A

n1

 am2 A

n 1

Therefore am   am1

 am 2 

n

n2

n2

 ......  a1 A

nm

0

 ......  a1

nm

 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

www.iosrjournals.org

43 | Page

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   an1

n1

n

 an2 n2  .....  a1  0 is the left   characteristic equation of A then

an m  an1m1  an2 m2  ......  a1mn  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 n2 such that n=m+1 then   characteristic equation of A is an   an1  an2   ......  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  c1r 1  c2 r 2  ......  cr 1 . Consider the following matrices:

B0  I

B1  A  c1 I

B2  A  c1A  c2 I 2

B3  A  c1 A  c2A  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  c1A

r 2

r 1

r

r

r 1

 c2A

r 2

 c3A

 c3 A

r 3

r 3

 ......  cr I ]

 ......  cr 1A]

  ( 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  c1r 1 I  c2 r 2 I  ......  cr 1I

 (r  c1r 1  c2 r 2  ......  cr 1 ) I  m  I The determinant on both sides gives www.iosrjournals.org

44 | Page

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    92 . Testing f   we have

and

is exactly only one of the following:

f  A  A2  9 A

www.iosrjournals.org

45 | Page

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.,

www.iosrjournals.org

46 | Page

F0634246