Vol 5, No 3 (2014)

Available online throug

ISSN 2229 – 5046

A NOTE ON RECTANGULAR MATRICES

Md. Shahidul Islam Khan*

Department of Mathematics, Nabajyoti College, Kalgachia, PIN-781319, India.

(Received on: 27-02-14; Revised & Accepted on: 15-03-14)

ABSTRACT

I

n this paper, we study about determinants, adjoint, inverse, 𝛼𝛼- characteristic equation, 𝛼𝛼-minimal polynomial and 𝛼𝛼-eigenvalues of rectangular matrices. We also prove some enlightening results.

1. INTRODUCTION

Let V be a

Γ

−

Banach algebra and m, n be positive integers. Denote by

V

_m,n and

Γ

_n,m the sets of

m

×

n

_matrices

with entries from V and

n

×

m

matrices with entries from

Γ

_{respectively. Let}

( ) ( )

x

_ij

,

y

_ij

∈

V

_m,n and

( )

γ

_ji

∈

Γ

_n,m, we define

( ) ( )( )( )

z

_ij

=

x

_ij

γ

_ji

y

_ij where _{ij ip pq qj}

p q

z

=

∑∑

x

γ

y

and

A

=

max

{

x

_ij

:

i

=

1, 2,.... ;

m j

=

1, 2,...

n

}

,

where

A

=

( )

x

_ij

∈

V

_m,n

.

Then

V

_m,n is a

Γ

_m,n

−

Banach algebra with respect to matrix addition, scalar multiplication, matrix multiplication and norm defined as above. The concept of

α

−

characteristic equations has been generalized by several authors [1], [2], [3], [5].In [3], define the product of rectangular matrices A and B of order

m

×

n

by

B

A

B

A

.

=

α

, for a fixed rectangular matrix

α

_n×m. With this product, we have

A

2

=

A

α

A

,

A

3

=

A

2

( )

α

A

,

( )

A

A

A

4

=

3

α

_,……..,

_A

n

=

_A

n−1

( )

α

_A

_.

2. DEFINITIONS AND EXAMPLES

Definition 2.1: A determinant

A

for a rectangular matrix

A

_m×_n

(

m

≤

n

)

Let

J

_n be the set of integers

{

1

,

2

,

3

,

...,

n

}

. Let the integers _{p p}m

p

k

k

k

m

,

1,

,

2

,

...

be such that

i.

m

≤

n

ii. _n

p

J

k

i

∈

for all

i

∈

J

_m and

p

=

1

,

2

,

3

,

...

iii. m

p p

p

k

k

k

₁,

<

2

<

...

<

For an integer d,

1

≤

d

≤

(

n

−

m

+

1

)

, define a set

S

_d such that

(

)

{

m

}

p p

p d

p

d

e

d

k

k

k

S

=

=

,

2

,

3

,

...,

If =n−d _m−1

d C

N , then the cardinal number of

S

_d is

N

_d .

The set

S

_d ,

1

≤

d

≤

(

n

−

m

+

1

)

will be ordered as follows. A set

S

_u

<

S

_v whenever

u

<

v

. Moreover, the elements

e

_pd and

e

_qd will be placed in the order

e

d_p

<

e

_qd whenever

k

_ps

<

k

_qs for s=2, 3… m.

All the m-tuples, therefore, admit of the following order; namely

1 1 1

1 2

2 2 1 1

2 1

1

...

1

...

2

...

.

...

..

+ −

+ − +

−

_<

_<

<

<

<

<

<

<

<

<

n m

m n m

n N

N

e

e

e

e

e

e

e

.

Corresponding author: Md. Shahidul Islam Khan*

Department of Mathematics, Nabajyoti College, Kalgachia, PIN-781319, India.

Consider the matrix

( )

n m ij

a

A= _× ,

m

≤

n

. Let d p

A be a sub-matrix of order m×m of A whose columns conform to the ordering of integers in d

p

e ;

1

≤

d

≤

(

n

−

m

+

1

)

,

1

≤

p

≤

N

_d

.

For an

m

×

n

(

m

≤

n

)

, matrix A with real elements. Let d

p

A be defined as above, then the number

( )

∑ ∑

+ − = = 1 1 1

det

m n d N p d p d

A

will be defined as the determinant of A and will be denoted by A.

For an

m

×

n

(

m

≥

n

)

, matrix A _{with real elements,} A will be defined as

A

T where

A

T denotes the transpose of matrix of A.

Example: Let _

     = 7 5 6 5 4 3 4 3 2 1

A . Here

m

=

2

and

n

=

5

, so 1≤d ≤

(

5−2+1

)

i.e. d =1,2,3, 4 and N1 =4, N2 =3, N3 =2, N4 =1. The sets S1,S2, S3and S4contains the following

elements, namely

( ) ( ) ( ) ( )

{

1,2, 1,3, 1,4, 1,5

}

1 =

S

( ) ( ) ( )

{

2,3, 2,4, 2,5

}

2 =

S

( ) ( )

{

3,4, 3,5

}

3 =

S

( )

{

4,5

}

4 =

S

There by the above definition, we have

7 5 6 5 4 3 4 3 2 1 = A 7 6 5 4 7 5 5 3 6 5 4 3 7 4 5 2 6 4 4 2 5 4 3 2 7 3 5 1 6 3 4 1 5 3 3 1 4 3 2 1 + + + + + + + + + =

=

−

₄₂

Definition 2.2: Cofactors, Adjoints and Inverse of rectangular matrix

Let

A

be a

m

×

n

(

m

≤

n

)

rectangular matrix of order

m

×

n

; then we have by definition that

A

is a linear homogeneous function of the entries in the ith row of A. If

c

_ij denotes the coefficient of

a

_ij

,

j

=

1

,

2

,

3

,

...,

n

, then we get the expression

in in i i i i i

i

c

a

c

a

c

a

c

a

A

=

_{1 1}

+

_{2 2}

+

_{3 3}

+

...

+

The coefficient

c

_ij of

a

_ij of the above expression is called the cofactor of

a

_ij. Let E, F, G and H be the sub-matrices of A of the order

(

i−1

) (

× j−1

) (

, i−1

) (

× n− j

)

,

(

m−i

) (

× n− j

)

and

(

m−i

) (

× j−1

)

respectively such

that           = G H a F E A ij : ... ... .... ... :

then the determinant of the sub-matrix

ij

M of the order

(

m−1

) (

× n−1

)

corresponds to the

G H

F E

Mij ₋

−

= , the cofactor of

ij

a , that is c_ij = M_ij .

A rectangular matrix A of order

m

×

n

(

m

≤

n

)

is said to be non-singular if

A

≠

0

; otherwise it is said to be singular.

If A is non-singular, then its inverse

A

−1 is defined by

A

1

Adj A

( )

.

A

−

₌

Definition 2.3:

α

−

_{characteristic Equation,}

α

−

_eigenvalue,

α

−

eigenvector

Let us consider a rectangular matrix

A

_m×n

(

m

≠

n

)

. Then we consider a fixed rectangular matrix

α

n×m of the

opposite order of

A

. Then

α

A

and

A

α

are both square matrices of order n and m respectively. If

m

<

n

, then

α

A

is the highest th

n order singular square matrix and

A

α

is the lowest th

m order square matrix forming their product. Then the matrix

α

A

−

λ

I

_n and

A

α

−

λ

I

_m are called the left

α

−

characteristic matrix and right

α

−

characteristic matrix of

A

respectively, where

λ

_{is an indeterminate. Also the determinant}

α

A

−

λ

I

_n is a polynomial in

λ

of degree n, called the left

α

−

characteristic polynomial of

A

and

A

α

−

λ

I

_m is a polynomial in

λ

of degree m, called the right

α

−

characteristic polynomial of

A

. That is, the characteristic polynomial of singular square matrix

A

α

is called the left

α

−

characteristic polynomial of

A

and the characteristic polynomial of the square matrix

A

α

is called the right

α

−

characteristic polynomial of

A

. The equations

α

A

−

λ

I

_n

=

0

and

A

α

−

λ

I

_m

=

0

are called the left

α

−

characteristic equation and right

α

−

characteristic equation of

A

respectively. Then the rectangular matrix

A

satisfies the left

α

−

characteristic equation, and the left

α

−

characteristic equation of

A

is called the

α

−

characteristic equation of

A

.

For, m>n the rectangular matrix

A

_m×_nsatisfies the right

α

−

characteristic equation of

A

. So in this case, the equation

0

=

−

I

m

A

α

λ

is called the

α

−

characteristic equation of

A

. The roots of the

α

−

characteristic equation of a rectangular matrix

A

are called the

α

−

eigenvalues of

A

If

λ

is an

α

−

eigenvalue of rectangular matrix

A

of order

m

×

n

(

m

<

n

)

,

then the matrix

α

A

−

λ

I

_n is singular. The equation

(

α

A

−

λ

I

_n

)

X

=

0

then possesses a non-zero solution i.e. there exists a non-zero column vector X such that

α

AX

=

λ

X

. A non-zero vector X satisfying this equation is called a

α

−

characteristic vector or

α

−

eigenvector of

A

corresponding to the

α

−

eigenvalue

λ

.

Definition 2.4:

α

−

minimal polynomial

For a rectangular matrix

A

_m×_n

(

m

≠

n

)

over a field K, let

J

( )

A

denote the collection of all polynomial

f

( )

λ

for which

f

( )

A

=

0

(Note that

J

( )

A

is non empty, since the

α

−

characteristic polynomial of A belongs to

J

( )

A

). Let

( )

λ

α

m

be the monic polynomial of minimal degree in

J

( )

A

. Then

m

_α

( )

λ

is called the

α

−

minimal polynomial of A.

3. MAIN RESULTS

Theorem 3.1: Let A be a rectangular matrix of order

m n

×

(

m

≤

n

)

.

i. If

m

=

1

, then

A

=

a

₁₁

+

a

₁₂

+

a

₁₃

+

...

+

a

_1n . ii. If

m

=

n

, then

A

=

det

( )

A

=

det

( )

A

₁1

iii. If any row of A is multiplied by c, then

A

is multiplied by c.

iv. If any two rows are identical, then

A

=

0

.

v. The values of a determinant changes in sign only, if any two rows are interchanged.

vi. If each element in a row is an algebraic sum of two or more quantities, then the determinant can be expressed as an algebraic sum of two or more determinants.

Proof: Proofs are straightforward and so omitted.

Theorem 3.2: If A is a rectangular matrix of order

m

×

n

(

m

≤

n

)

_{, then}

i.

( )

m

I A A Adj

A. = where I_m is the unit matrix of order m. ii. Adj

( )

A.A=a singular matrix of order n.

Theorem 3.3: If A is an

m

×

n

(

m

<

n

)

rectangular matrix and

α

is an

n

×

m

rectangular matrix, then A is a zero of its

α

−

characteristic polynomial.

Proof: Since A is an

m

×

n

(

m

<

n

)

rectangular matrix and

α

is an

n

×

m

rectangular matrix, so the

α

−

characteristic polynomial of A is of the form

( )

n n m

m n m n

m

a

a

a

a

A

I

−

=

+

₋ −

+

₋ −

+

+

−

=

∆

λ

λ

α

λ

λ

λ

2 ₀

λ

2 1

1

...

.

Let B

( )

λ denote the classical adjoint of the matrix

λ

I

−

α

A

. The elements of B

( )

λ are cofactors of the matrix

A

I

α

λ

−

and hence are polynomials in

λ

of degree not exceeding

n

−

1

.

Thus

B

( )

λ

=

B

_m−

λ

n−

+

B

_m−

λ

n−

+

+

B

n−m+

+

B

0

λ

n−m 1

1 2

2 1

1

...

_, where the Bi are n-square matrices over K which

are independent on

λ

. By the fundamental property of the adjoint

(

λI −αA

) ( )

B λ = λI −αAI

(

)(

n n m n m

)

m n

m

B

B

B

B

A

I

−

α

₋

λ

−

+

₋

λ

−

+

+

− +

+

λ

−

λ

1 ₀

1 2

2 1

1

...

(

a

a

a

a

)

I

m n n m n m n m − − − − −

+

+

+

+

=

λ

λ

λ

2 ₀

λ

2 1

1

...

Removing parentheses and equating the coefficients of corresponding of

λ

_,

I a B_m−1 = _m

( )

A B a I

Bm−2 −

α

m−1 = m−1

( )

A B a I

B_m−3 −

α

_m−2 = _m−2 ……….

……….

( )

A B a I

B₀ −

α

₁ = ₁ −

( )

α

A B0 =a0

Multiplying the above matrix equations by An, An−1, An−2,..., An−m+1, An−m respectively.

n m m n A a B A ₋₁ =

1 1 1 2 1 − − − −

− _{− =} n

m m n m n A a B A B A 2 1 2 1 3 2 − − − − −

− _{− =} n

m m n m n A a B A B A ……… ……… 1 1 1 2 0

1 − + − +

+

−m ₋ n m ₌ n m n A a B A B A

_{− A}n−m+ _{B =a A}n−m

0 0

1

Adding the above matrix equations, we get

m n n m n m n

m

A

a

A

a

A

a

A

a

+

₋ −

+

₋ −

+

+

−

=

0 2 2 1 1

...

0

In other words, ∆

( )

A =0. That A is a zero of its characteristic polynomial.

Theorem 3.4: Let A and

α

be two rectangular matrices of order

m

×

n

and

m

n

×

(

m

<

n

)

respectively. If

m

_α

( )

λ

=

λ

r

+

c

₁

λ

r−1

+

c

₂

λ

r−2

+

...

+

c

_r−1

λ

is the

α

−

minimal polynomial of

A and

B

_r

=

( )

α

A

r

+

c

₁

( )

α

A

r−1

+

c

₂

( )

α

A

r−2

+

c

₃

( )

α

A

r−3

+

...

+

c

_r

I

, then

−

α

AB

r−1

=

0

.

Proof: Suppose

m

_α

( )

λ

=

λ

r

+

c

₁

λ

r−1

+

c

₂

λ

r−2

+

...

+

c

_r−1

λ

. Consider the following matrices:

I

B

₀

=

I

c

A

B

1

=

α

+

1

( )

A

c

A

c

I

( )

A

c

( )

A

c

A

c

I

B

₃

=

α

3

+

₁

α

2

+

₂

α

+

₃

……… ………

( )

A

c

( )

A

c

( )

A

c

I

B

_r−1

=

α

r−1

+

₁

α

r−2

+

₂

α

r−3

+

...

+

_r−1

( )

A

c

( )

A

c

( )

A

c

( )

A

c

I

B

_r

=

α

r

+

₁

α

r−1

+

₂

α

r−2

+

₃

α

r−3

+

...

+

_r

Then

B

0

=

I

I

c

AB

B

1

−

α

0

=

1

I

c

AB

B

2

−

α

1

=

2

I

c

AB

B

3

−

α

2

=

3

……….. ………..

I

c

AB

B

_r−₁

−

α

_r−₂

=

_r−₁

Thus

−

α

AB

_r−1

=

c

_r

I

−

B

_r

=

c

_r

I

−

[

( )

α

A

r

+

c

₁

( )

α

A

r−1

+

c

₂

( )

α

A

r−2

+

c

₃

( )

α

A

r−3

+

...

c

_r

I

]

[

...

1

]

3

3 2 2 1

1

A

c

A

c

A

c

A

c

A

r

α

r

α

r

α

r _r

α

α

+

−

+

−

+

−

+

−

−

=

=

−

α

(

A

r

+

c

₁

A

r−1

+

c

₂

A

r−2

+

c

₃

A

r−3

+

...

c

_r−1

A

)

=

−

α

.

m

_α

( )

A

=

−

α

.

0

=

0

REFERENCES

[1]. G. L. Booth, “Radicals of matrix rings”, Math. Japonica, Vol.33, 1988, pp. 325-334.

[2]. 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 ring over ring”, M. J. Xinjiang Univ. Nat. Sci., Vol. 4, 1987, pp. 37-42.

[4]. P. Rajkhowa and Md. Shahidul Islam Khan, “On α-Characteristic Equations and α-Minimal Polynomial of Rectangular Matrices”, IOSR J. of Mathematics, Vol. 6, Issue 3, 2013, pp. 42- 46.

[5]. T. K. Dutta and H. K. Nath, “On the Gamma ring of rectangular Matrices”, Bulletin of Pure and Applied Science, Vol. 16E, 1997, pp. 207-216.

Source of support: Nil, Conflict of interest: None Declared

−

Γ

−