on The Hadamard Products of Its Adjoint Matrix With a Square Matrix

S. Ü. Fen-Edebiyat Fakültesi Fen Dergisi Sayı 17[2000]43-49,KONYA

on The Hadamard Products of Its Adjoint Matrix With

a Square Matrix

Dursun TAŞCI 1

Abstract: In this paper for any matrix A=(a_ij)∈M_n(R) we defined ϕ(A)=AοadjA and

(

)

T

T(A)=Aο adjA

ϕ , where T denotes transpose and

ο

denotes Hadamard product. We obtained some properties of ϕ(A) and ϕ_T(A).

Key Words: Hadamard product, Adjoint matrix

Bir Kare Matris ile Onun Adjoint Matrisinin Hadamard

Çarpımları Üzerine

Özet: Bu çalışmada reel elemanlı herhangi bir n×n A kare matrisi göz önüne alınarak

ve adjA A ) A ( = ο ϕ

(

)

T T(A)=Aο adjA

ϕ matrisleri tanımlandı ve bu matrislerin bazı özellikleri elde edildi.

Anahtar Kelimeler: Hadamard Çarpımı, Adjoint Matris.

Introduction and the Main Results

Firstly we give the following definitions

Definition 1.[1] The Hadamard product of A =(a_ij)∈M_n and B=(b_ij)∈M_n is defined by

. n ij ijb ) M a ( B Aο = ∈

Definition 2.[2] Let be an n×n matrix over any commutative ring. The permanent of A, written , is defined by ) a ( A= _ij ) A ( per

∑

σ σ σ σ = a ₍₎a _{( )} a_{n (n)} ) A ( per _{1 1 2 2 K} ,

where the summation extends over all one-to-one functions from

{

1

,

2

,

_K

,

n

}

to

{

1

,

2

,

_K

,

n

}

.

D.TAŞCI

Definition 3.[3] For any matrix A =(a_ij)∈M_n the column matrix norm 1

.

is defined by

∑

= ≤ ≤ = n i ij n j a max A 1 1 1 .

Definition 4.[3] For any matrix A =(a_ij)∈M_n the row matrix norm

.

_∞ is defined by

∑

= ≤ ≤ ∞ = n j ij n i a max A 1 1 .

Definition 5.[3] For any

A

=

(

a

_ij

)

∈

M

_n, the Euclidean matrix norm is defined by

2 1 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑

= n j ,i ij E a A .

Definition 6.[3] For any matrix

A

=

(

a

_ij

)

∈

M

_n sum matrix norm t

.

is defined by

∑

= = n j, i ij t a A 1 .

Now we present the main results.

Theorem 1. For any A=(a_ij)∈M₂(R)

(

(A)

)

det(A)per(A)

det ϕ =

and

(

(A)

)

det(A)per(A)

detϕ_T = ,

where

M

₂

(

R

)

denotes 2×2 matrices with real entries, ϕ(A)=AοadjA and .

(

)

T T(A)=Aο adjA

ϕ

Proof. For any matrix

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = 22 21 12 11 a a a a A since ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = 11 21 12 22 a a a a adjA we write ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − = ο = ϕ 22 11 2 21 2 12 22 11 a a ) a ( ) a ( a a adjA A ) A ( .

ON THE HADAMARD PRODUCTS OF ITS ADJOINT MATRIX WITH A SQUARE MATRIX

If we compute the determinant of ϕ(A), then we find

(

)

. ) A ( per ) A det( ) a a a a ( ) a a a a ( ) a a ( ) a a ( ) A ( det = + − = − = ϕ 21 12 22 11 21 12 22 11 2 21 12 2 22 11

Similarly it is easily seen that

(

(A)

)

det(A)per(A)

detϕ_T = .

Thus the proof is complete.

Remark 1. Unfortunately Theorem 1 is wrong for . Let us give a counterexample.

Consider

n

3

≥

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 4 2 2 1 2 1 3 2 1 A . Since ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 0 2 2 2 2 2 4 2 6 A adj we find

ϕ

. ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 0 4 4 2 4 2 12 4 6 ) A (

On the other hand, since det

(

ϕ )(A

)

=272, det

(

ϕ_T(A)

)

=216, det(A)=−4 and , it follows that

40

perA

=

(

(

)

)

det(

)

(

)

det

ϕ

A

≠

A

per

A

and

(

(A)

)

det(A)per(A) det ϕT ≠ .

Theorem 2. For any matrix A=(a_ij)∈M₂(R), the eigenvalues of are , . Also the eigenvalues of are

)

A

(

ϕ

λ₁=detA ) A ( per = λ₂ ϕ_T(A)

λ

₁

=

det

A

, λ₂ =per(A).

Proof. For any matrix

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 22 21 12 11 a a a a A since ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − = ϕ 22 11 2 21 2 12 22 11 a a ) a ( ) a ( a a ) A (

D.TAŞCI we have

(

) (

)

2 21 12 2 22 11 22 11 2 22 11 2 21 2 12 22 11 2a a a a a a a a ) a ( ) a ( a a det )) A ( I det( − + λ − λ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − λ − λ = ϕ − λ (1.1)

On the other hand the roots of the equation (1.1) are 21 12 22 11 1 =a a −a a λ and λ₂ =a₁₁a₂₂ +a₁₂a₂₁. Moreover since 21 12 22 11a a a a A

det = − and per(A)=a₁₁a₂₂ +a₁₂a₂₁

we find

A det =

λ₁ and λ₂ =per(A).

Similarly it is easily seen that the eigenvalues of ϕ_T(A) are λ₁ =detA and λ₂ =per(A). So the theorem is proved.

Remark 2. Unfortunately Theorem 2 is wrong for . But is always an eigenvalue

of for . We state this fact as theorem .

3

n

≥

det

A

) A ( T ϕ

n

≥

3

Theorem 3. For

n

≥

3

, at least an eigenvalue of ϕ_T(A) is equal to detA.

Proof. We remark that

in in i i i i a a a A det = 1α1+ 2α2 +K + α ,

(

i

=

1

,

2

,

K

,

n

)

(1.2) or nj nj j j j j a a a A det = ₁ α₁ + ₂ α₂ +_K + α ,

(

j

=

1

,

2

,

_K

,

n

)

(1.3) where

α

_ij denotes cofactor of

a

_ij, i.e., α_ij =(−1)i+jdet(A_ij).

Using the properties of determinants and considering (1.2) we have

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ α − λ α − α − α − α − λ α − α − α − α − λ = ϕ − λ nn nn n n n n n n n n T a a a a a a a a a det )) A ( I det( K M M M K K 2 2 1 1 2 2 22 22 21 21 1 1 12 12 11 11

ON THE HADAMARD PRODUCTS OF ITS ADJOINT MATRIX WITH A SQUARE MATRIX ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ α − λ α − α − λ α − α − λ α − λ α − α − α − λ =

∑

∑

∑

= = = nn nn n n n j nj nj n n n j j j n n n j j j a a a a a a a a a det K M M M K K 2 2 1 2 2 22 22 1 2 2 1 1 12 12 1 1 1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ α − λ α − − λ α − α − λ − λ α − α − − λ = nn nn n n n n n n a a A det a a A det a a A det det K M M M K K 2 2 2 2 22 22 1 1 12 12 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ α − λ α − α − α − λ α − α − − λ = nn nn n n n n n n a a a a a a det ) A det ( K M M M K K 2 2 2 2 22 22 1 1 12 12 1 1 1 ,

it follows that detA is eigenvalue of ϕ_T(A).

Theorem 4. Let A be n×n real matrix, then all the row sums and column sums of are equal to . ) A ( T ϕ A det

Proof. If

r

₁

,

r

₂

,

_K

,

r

_n denote the row sums of

ϕ

_T

(

A

)

, then we write

∑

= α = n j ij ij i a r 1 . On the other hand considering (1.2) we have

∑

= α = = n j ij ij i a r A det 1

(

i

=

1

,

2

,

_K

,

n

)

.

Similarly if

c

₁

,

c

₂

,

_K

,

c

_n denote the column sums of

ϕ

_T

(

A

)

, then we write

∑

= α = n i ij ij j a c 1

D.TAŞCI

Again considering (1.3) we have

∑

= α = = n i ij ij j a c A det 1

(

j

=

1

,

2

,

_K

,

n

)

. Thus the proof is complete.

Corollary 1. For A =(a_ij)∈M_n(R), A det n e ) A ( eTϕ_T = , where e=(1,1,_K,1)T . Proof. We write

∑ ∑

∑

= = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ α = α = ϕ n i n j ij ij n j, i ij ij T T _{(A)e a a} e 1 1 1 A det n A det n i = =

∑

=1

and the proof is complete.

Theorem 5. For any matrix

A

=

(

a

_ij

)

∈

M

_n

(

R

)

, the following statements are true. (i) ϕ_T(A) ≥ detA

1

(ii) ϕ_T(A)_∞ ≥ detA

(iii) ϕ_T(A) _E ≥ detA

(iv) ϕ_T(A)_t ≥n detA

Proof. (i) Considering the definition 3 and triangle inequality we have

A det A det max a max a max ) A ( n j n i ij ij n j n i ij ij n j T = α ≥ α = = ϕ ≤ ≤ = ≤ ≤ = ≤ ≤

∑

₁ 1

∑

₁ 1 1 1 .

(ii) By t he definition 4 and triangle inequality, we write?? EMBED Equation.3??

? ?

.

? (iii) Considering the definition 5 and triangle inequality

EMBED Equation.3??

? it follows that??

EMBED Equation.3??

?? ON THE HADAMARD PRODUCTS OF ITS ADJOINT MATRIX WITH

(iv) Similarly by the definition 6 and triangle inequality we have?? EMBED Equation.3??

? ?

Horn, R.A., Johnson, C:R., Topics in Matrix Analysis, Cambridge University Press. (1991)

Minc, H., Permanents, In Encyclopaedia of Mathematics and ItsApplications, Vol. 6, Addison-Wesley (1978)