division of matrices and mirror image ......rows of matrices j,k and s are equal to the columns of...

DIVISION OF MATRICES AND MIRROR IMAGE PROPERTIES OF MATRICES

Neelam Jeevan Kumar Electric and Electronics Engineering,

H.No: 19-6-194, Rangashaipet, Warangal, Andhra Pradesh, India-506005 Contact No: + 91 9492907696

Email: [email protected] In Algebra, Division of matrices is done by Inverse-Multiplication Divisor Matrix with Dividend Matrix. There are two different methods are possible to divide two square or rectangular matrices. The Probability of Matrices Division is 0.5 plus. The overall concept of this paper is Division of Matrices is possible. Key Words: Law Matrices Multiplication[a]; Slash and Back Slash Matrices/Mirror Image Matrix[b]; Matrices Commutative Law[c]. Matrix Digonalization[d]; Product Matrix[e]

1. Introduction

Let us consider Two matrices J and K having m number of rows and n number of columns and the product of matrices J and K is S

퐉 =

⎣⎢⎢⎢⎡퐣ퟏퟏ 퐣ퟏퟐ… . 퐣ퟏ퐧퐣ퟐퟏ 퐣ퟐퟐ… . 퐣ퟐ퐧: ∶ ∶: ∶ ∶

퐣퐦ퟏ 퐣퐦ퟐ… . 퐣퐦퐧⎦⎥⎥⎥⎤

퐊 =

⎣⎢⎢⎢⎡퐤ퟏퟏ 퐤ퟏퟐ… .퐤ퟏ퐧퐤ퟐퟏ 퐤ퟐퟐ… .퐤ퟐ퐧

: ∶ ∶: ∶ ∶

퐤퐦ퟏ 퐤퐦ퟐ… .퐤퐦퐧⎦⎥⎥⎥⎤

jmxjn kmxkn

Where m ≠ n for Rectangular matrices and

m = n for Square matrices

The product of matrices J and K is matrix-S called Product Matrix[d]

(i.e., jn = km)

[ J ] [ K ] = [ S ] ………. ( i )

⎣⎢⎢⎢⎡퐣ퟏퟏ 퐣ퟏퟐ… . 퐣ퟏ퐧퐣ퟐퟏ 퐣ퟐퟐ… . 퐣ퟐ퐧: ∶ ∶: ∶ ∶

퐣퐦ퟏ 퐣퐦ퟐ … . 퐣퐦퐧⎦⎥⎥⎥⎤

⎣⎢⎢⎢⎡퐤ퟏퟏ 퐤ퟏퟐ… .퐤ퟏ퐧퐤ퟐퟏ 퐤ퟐퟐ… .퐤ퟐ퐧

: ∶ ∶: ∶ ∶

퐤퐦ퟏ 퐤퐦ퟐ… . 퐤퐦퐧⎦⎥⎥⎥⎤

=

⎣⎢⎢⎢⎡퐬ퟏퟏ 퐬ퟏퟐ… . 퐬ퟏ퐧퐬ퟐퟏ 퐬ퟐퟐ… . 퐬ퟐ퐧

: ∶ ∶: ∶ ∶

퐬퐦ퟏ 퐬퐦ퟐ… . 퐬퐦퐧⎦⎥⎥⎥⎤

General Condition to divide two matrices

1. Determinant of Divisor Should not be Zero (i.e., |A|/|B| = element-c. |B| ≠ 0 ) 2. Either divisor row/column must be equal to dividend row/column

International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013 ISSN 2278-7763

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Copyright © 2013 SciResPub. IJOART

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Proof: for condition -2

Multiplicand is unknown i.e., [J] is unknown.

To find order matrix –J = [ J ]

[ J ] = [ S ] / [ K ]

JM X JN = (SM X SN) / (KM X KN) = (JM X KN) / (KM X

KN)

In this case Dividend and Divisor columns must be

Equal to get Quotient (i.e.,[ J ]).

The order of Quotient matrix is

dividend row x divisor row

Multiplier is unknown i.e., [ K ] is unknown.

To find order matrix –K =[ K ]

[ K ] = [ S ]/[ J ]

KM X KN= (SM X SN) / (JM X JN) = (JM X KN) / (JM X

JN)

In this case Dividend and Divisor rows must be Equal

to get Quotient(i.e.,[ K ]).

The order of Quotient matrix is

divisor column x dividend column

2. Methods

i) Determinants ii) Solving Linear Equations

Method 2.i. used for Square matrices (i.e., m = n) and method 2.ii. Is used for both Rectangular and Square matrices

2.i. Division by Determinants

Rows of Matrices J,K and S are equal to the Columns of Matrices J,K and S

This method is based on Cramer’s Rule

Jm = Jn : Km = Kn : Sm = Sn

2.i.a. When Rows of Dividend & Divisor are Equal the elements of [ J ]

퐉퐬퐦, 퐬퐦 → 퐤퐦 = [푲]풔풎,풔풎 → 풌풎− −−− −−−

[푲]풌풎,풌풏 … ( 풊풊 )

Here sm→km means, Substituting/Replacing mth row of

[S] in mth row of [K]

( Determinant of replaced or substituted matrix by rows

) / ( Determinant of actual/original/primary matrix )

2.i.b when Columns of Dividend & Divisor are Equal

the elements of [ K ]

퐊퐬퐧 → 퐣퐧, 퐬퐧 =[푱]풔풏 → 풋풏, 풔풏−−− −− −−

[푱]풋풎, 풋풏 … . ( 풊풊풊 )

Here sm→jn means by Substituting/Replacing nth

column of [S] in nth column of [J]

(Determinant of replaced or substituted matrix by

column) / (Determinant of actual/original/primary

matrix)


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2.ii. Division by solving Linear equations

Rows of Matrices J,K and S are not equal to the Columns of Matrices J,K and S

Jm ≠ Jn : Km ≠ Kn : Sm ≠ Sn

This method is based on Gauss Seidel method

2.ii.a when Rows of Dividend & Divisor are Equal

⎣⎢⎢⎢⎡풙ퟏퟏ 풙ퟏퟐ… .풙ퟏ풏풙ퟐퟏ 풙ퟐퟐ… .풙ퟐ풏

: ∶ ∶: ∶ ∶

풙풎ퟏ 풙풎ퟐ… . 풙풎풏⎦⎥⎥⎥⎤

⎣⎢⎢⎢⎡풌ퟏퟏ 풌ퟏퟐ… .풌ퟏ풏풌ퟐퟏ 풌ퟐퟐ… .풌ퟐ풏

: ∶ ∶: ∶ ∶

풌풎ퟏ 풌풎ퟐ… . 풌풎풏⎦⎥⎥⎥⎤

=

⎣⎢⎢⎢⎡풔ퟏퟏ 풔ퟏퟐ… . 풔ퟏ풏풔ퟐퟏ 풔ퟐퟐ… . 풔ퟐ풏

: ∶ ∶: ∶ ∶

풔풎ퟏ 풔풎ퟐ… . 풔풎풏⎦⎥⎥⎥⎤

[ K ] and [ S ] are known matrices. [ X ] is unknown

matrix

Now solve the unknown elements of unknown matrix.

x11k11 + x12k21 + x13k31 +…………+ x1nkm1 = s11

x11k12 + x12k22 + x13k32 +…………+ x1nkm2 = s12

x11k13 + x12k23+ x13k33 +…………+ x1nkm3 = s13

to

x11k1n + x12k2n + x13k3n +…………+ x1nkmn = s1n

Find x11 to x1n values.

And repeat same procedure to find remaining

x21 to x2n values

x31 to x3n values

to

xm1 to xmn values final matrix

substitute the values in matrix-X which is equal to

Matrix-J

⎣⎢⎢⎢⎡풙ퟏퟏ 풙ퟏퟐ… . 풙ퟏ풏풙ퟐퟏ 풙ퟐퟐ… . 풙ퟐ풏

: ∶ ∶: ∶ ∶


=

⎣⎢⎢⎢⎡풋ퟏퟏ 풋ퟏퟐ… . 풋ퟏ풏풋ퟐퟏ 풋ퟐퟐ… . 풋ퟐ풏

: ∶ ∶: ∶ ∶

풋풎ퟏ 풋풎ퟐ… . 풋풎풏⎦⎥⎥⎥⎤

2.ii.b when Columns of Dividend & Divisor are Equal

⎣⎢⎢⎢⎡풋ퟏퟏ 풋ퟏퟐ… . 풋ퟏ풏풋ퟐퟏ 풋ퟐퟐ… . 풋ퟐ풏

: ∶ ∶: ∶ ∶

풋풎ퟏ 풋풎ퟐ… . 풋풎풏⎦⎥⎥⎥⎤

⎣⎢⎢⎢⎡풙ퟏퟏ 풙ퟏퟐ… .풙ퟏ풏풙ퟐퟏ 풙ퟐퟐ… .풙ퟐ풏

: ∶ ∶: ∶ ∶


=

⎣⎢⎢⎢⎡풔ퟏퟏ 풔ퟏퟐ… . 풔ퟏ풏풔ퟐퟏ 풔ퟐퟐ… . 풔ퟐ풏

: ∶ ∶: ∶ ∶

풔풎ퟏ 풔풎ퟐ… . 풔풎풏⎦⎥⎥⎥⎤

[ J ] and [ S ] are known matrices. [ X ] is unknown

matrix.

Now solve the unknown elements of unknown matrix.

j11x11 + j12x21 + j13x31 +…….+ j1nxm1 = s11

j21x11 + j22x21 + j23x31 +…….+ j2nxm1 = s21

j31x11 + j32x21 + j33x31 +…….+ j3nxm1 = s31

to

jm1x11 + jm2x21 + jm3x31 +…….+ jmnxm1 = sm1

Find x11 to x1n values

And repeat same procedure to find remaining

x21 to x2n values

x31 to x3n values

to

xm1 to xmn values

substitute the values in matrix-X which is equal to

Matrix-K

⎣⎢⎢⎢⎡풙ퟏퟏ 풙ퟏퟐ… . 풙ퟏ풏풙ퟐퟏ 풙ퟐퟐ… . 풙ퟐ풏

: ∶ ∶: ∶ ∶

풙풎ퟏ 풙풎ퟐ… .풙풎풏⎦⎥⎥⎥⎤

=

⎣⎢⎢⎢⎡풌ퟏퟏ 풌ퟏퟐ… .풌ퟏ풏풌ퟐퟏ 풌ퟐퟐ… .풌ퟐ풏

: ∶ ∶: ∶ ∶

풌풎ퟏ 풌풎ퟐ… .풌풎풏⎦⎥⎥⎥⎤


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3. Mirror Image Matrix[b] In mirrors Reflection image is opposite to Real image. Mirror shows Right as left and vice versa.

Figure 1: Matrix-A. = [A.] is Mirror image matrix of matrix-A = [ A] and Matrix-B. = [B.] is Mirror image matrix of matrix-B = [ B]

Example: [ A ] [ B ] = [ B ] [A. ] = [ B. ][ A] [ C] ….. ( iv )

Properties if Mirror Image Matrices

3.i Law of Matrix Multiplication

The Product of sum of Multiplicand Matrix nth column elements and sum of Multiplier Matrix mth row elements is

equal to the sum of product Matrix elements

∑ 푪풋푹풌푱풏 푲풎풎 풏 ퟏ = ∑ 푺풎풏

풎 푱풎풏 푲풏풎 ퟏ,풏 ퟏ

….. ( v )

Where Rk is sum of Row elements of matrix-k and

Cj is sum of Column elements of Matrix-j

3.ii Eigen Values

Let matrix - A. = [ A. ] is the Mirror Image Matrix of Matrix – A = [ A ]

[ A. ] =

⎣⎢⎢⎢⎡풂.ퟏퟏ 풂.ퟏퟐ… .풂.ퟏ풏풂.ퟐퟏ 풂.ퟐퟐ… .풂.ퟐ풏

: ∶ ∶: ∶ ∶

풂.풎ퟏ 풂.풎ퟐ… .풂.풎풏⎦⎥⎥⎥⎤ and [ A ] =

⎣⎢⎢⎢⎡풂ퟏퟏ 풂ퟏퟐ… .풂ퟏ풏풂ퟐퟏ 풂ퟐퟐ… .풂ퟐ풏

: ∶ ∶: ∶ ∶

풂풎ퟏ 풂풎ퟐ… .풂풎풏⎦⎥⎥⎥⎤

A. x = λ x ….. ( vi .a )

A y = λ y …... ( vi. b )

[ C] = [B.][A]

Mirror

[ C] = [B.][A] [ A ][ B] = [C]

[ C] = [B][A.]


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The Characteristic equations of both [ A ] and [ A. ] are same because They obeys Mirror image

property and λ (Eigen value column matrix) is same

But Eigen Vector are Different because elements of Both [ A ][ A. ] are not same i.e., [ x ] ≠ [ y ]

|푨.− 흀푰| = |푨 − 흀푰 | = ퟎ …….. ( vi. c )

3. iii Trace


[ A. ] =


: ∶ ∶: ∶ ∶

풂.풎ퟏ 풂.풎ퟐ… .풂.풎풏⎦⎥⎥⎥⎤ and [ A ] =


: ∶ ∶: ∶ ∶


Definition: Trace is the sum of Diagonal Elements of a Matrix

Trace ( A. ) = ∑ 풂.풑풑풏풑 ퟏ = a.11 + a.22 + a.33 + a.44 + ….. + a.mn ……….. ( vii. a)

Trace ( A ) = ∑ 풂풑풑풏풑 ퟏ = a11 + a22 + a33 + a44 + ….. + amn ……….. ( vii. b )

Matrices A. is Mirror image to A

( vii. a ) = ( vii. b )

Trace ( A ) = Trace ( A. )

∑ 풂.풑풑풏풑 ퟏ = ∑ 풂풑풑풏

풑 ퟏ

3.iv Matrix Digonalization[d]


[ A. ] =


: ∶ ∶: ∶ ∶

풂.풎ퟏ 풂.풎ퟐ… .풂.풎풏⎦⎥⎥⎥⎤ and [ A ] =


: ∶ ∶: ∶ ∶


P-1 A P = D1 ………….. ( viii .a )

Q-1 A. Q = D2 ………….. ( viii .b )

Where P and Q are Block Matrices of [ A ] and [ A. ] respectively

But [ A. ] is Mirror Inage to Matrix [ A ]

Diagonalized elements Must be Same i.e., D1 = D2 = D


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4 Example

For Square Matrices Let [A] = ퟏ ퟐ

ퟑ ퟒ [B]= ퟓ ퟔퟕ ퟖ are 2x2 matrices

respectively. The product of [ A ] [B ] = [ C ] = ퟏퟗ ퟐퟐ

ퟒퟑ ퟓퟎ

Determinant of matrices A and B are -2 and -2

| 퐴 | = | 퐵 | = −2

i) Assuming matrix-A is unknown

[ A ] = [ C ] / [ B ] = ( 2 x 2 ) / ( 2 x 2 ) i.a) Rows are Equal (i.e., Km = Sm)

퐊퐬퐧 → 퐣퐧, 퐬퐧 =[푱]풔풏 → 풋풏, 풔풏−−− −− −−

[푱]풋풎, 풋풏

a11=ퟏퟗ ퟔퟒퟑ ퟖ

ퟐ = -106/-2 = 53 : a12 =

ퟐퟐ ퟔퟓퟎ ퟖ

ퟐ = -124/-2 = 62

a21 =ퟓ ퟏퟗퟕ ퟒퟑ

ퟐ = 82/-2= - 41 : a22 =

ퟓ ퟐퟐퟕ ퟓퟎ

ퟐ = 96/-2 = -48

[ A.] = 퐚ퟏퟏ 퐚ퟏퟐ퐚ퟐퟏ 퐚ퟐퟐ = ퟓퟑ ퟔퟐ

−ퟒퟏ −ퟒퟖ i.b) Columns are Equal (i.e., Kn = Sn)

퐉퐬퐦, 퐬퐦 → 퐤퐦 = [푲]풔풎, 풔풎 → 풌풎−−− −− −−

[푲]풌풎,풌풏

a11=ퟏퟗ ퟐퟐퟕ ퟖ

ퟐ = -2/-2 = 1 : a12=

ퟓ ퟔퟏퟗ ퟐퟐ

ퟐ = -4/-2 = 2

a21=ퟒퟑ ퟓퟎퟕ ퟖ

ퟐ = -6/-2 = 3 : a22=

ퟓ ퟔퟒퟑ ퟓퟎ

ퟐ= -8/-2 = 4

[ 퐀 ] = 퐚ퟏퟏ 퐚ퟏퟐ퐚ퟐퟏ 퐚ퟐퟐ = ퟏ ퟐ

ퟑ ퟒ

[ A ] [ B ] = [ B ] [A. ] = [ C] [A] and [A.] are follows Mirror image matrix[b]

i) Assuming matrix-B is unknown

[ B ] = [ C ] / [ A ] = ( 2 x 2 ) / ( 2 x 2)

ii.a) Rows are Equal (i.e., Km = Sm)

퐊퐬퐧 → 퐣퐧, 퐬퐧 =[푱]풔풏 → 풋풏, 풔풏−−− −− −−

[푱]풋풎, 풋풏

b11=ퟏퟗ ퟐퟒퟑ ퟒ

ퟐ = -10/-2 = 5 : a12 =

ퟐퟐ ퟐퟓퟎ ퟒ

ퟐ = -88/-2 = 6

b21 =ퟏ ퟏퟗퟑ ퟒퟑ

ퟐ = -14/-2=7 : a22 =

ퟏ ퟐퟐퟑ ퟓퟎ

ퟐ = -16/-2 = 8

[ B] = 퐛ퟏퟏ 퐛ퟏퟐ퐛ퟐퟏ 퐛ퟐퟐ = ퟓ ퟔ

ퟕ ퟖ ii.b) Columns are Equal (i.e., Kn = Sn)

퐉퐬퐦, 퐬퐦 → 퐤퐦 = [푲]풔풎, 풔풎 → 풌풎−−− −− −−

[푲]풌풎,풌풏

a11=ퟏퟗ ퟐퟐퟑ ퟒ

ퟐ = 10/-2 = -5 : a12=

ퟏ ퟐퟏퟗ ퟐퟐ

ퟐ = -16/-2 = 8

a21=ퟒퟑ ퟓퟎퟑ ퟒ

ퟐ = 22/-2 = -11 : a22=

ퟏ ퟐퟒퟑ ퟓퟎ

ퟐ= -36/-2 = 18

[ 퐁 ] = 퐛ퟏퟏ 퐛ퟏퟐ퐛ퟐퟏ 퐛ퟐퟐ = −ퟓ ퟖ

−ퟏퟏ ퟏퟖ

[ A ][ B ] = [ B. ][ A] = [ C ] [ B ] and [ B.] are follows Mirror image matrix[b]

Law of Matrix Multiplication

[ A ] = ퟏ ퟐퟑ ퟒ [ B ] = ퟓ ퟔ

ퟕ ퟖ

[ A ][ B ] = [ C ] = ퟏퟗ ퟐퟐퟒퟑ ퟓퟎ

(1+3)(5+6)+(2+4)(7+8)=(19+22+43+50)=134


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[ B ][ A. ] = [ C ] ( 5+7 )( 53+62 )+( 6+8 )( -41-48 )=( 19+22+43+50 )=134

[ B. ][ A] = [ C ] (-5-11 )( 2+1 )+( 18+8 )( 3+4 ) = ( 19+22+43+50 )=134 Trace Tr( A ) = sum of Diagonal elements of Matrix - A Tr( A ) = 1+4 = 5 Tr( A. ) = 53-48 = 5

Tr ( A ) = Tr ( A. ) Tr ( B ) = 5 + 8 = 13 Tr ( B.) = -5 + 18 = 13

Tr ( B ) = Tr ( B. ) From the Calculated Traces Traces of [A] and [A.] are Equal and Traces of [B] and

[B.] are Equal

By the law of Matrix Multiplication 5.c Eigen Values

Eigen Vales of Matrix-J

| λI − A | = 0 흀 − ퟏ −ퟐ−ퟑ 흀 − ퟒ = 0 휆2 - 5λ – 2 = 0

λ1 = -0.3372281323269, λ2 = 5.372281323269

Eigen Value of Matrix- A.

| λI − A. | = 0 흀 − ퟓퟑ −ퟔퟐퟒퟏ 흀 + ퟒퟖ = 0 휆2 - 5λ – 2 = 0

λ1 = -0.3372281323269, λ2 = 5.372281323269

Eigen Vales of Matrix-B

| λI − K | = 0

흀 − ퟓ −ퟔ−ퟕ 흀 − ퟖ = 0 휆2 - 13λ – 2 = 0

λ1 = 13.152067347825, λ2 = -0.152067347825

Eigen Vales of Matrix-B.

| λI − K. | = 0 흀 + ퟓ −ퟖퟏퟏ 흀 − ퟏퟖ = 0 휆2 - 13λ – 2 = 0

λ1 = 13.152067347825, λ2 = -0.152067347825 From the Calculated Eigen Values Eigen Values of [A] and [A.] are Equal and Eigen

Values of [B] and [B.] are Equal.

Matrix Digonalization [ A ] = [ A. ] = −ퟎ.ퟑퟕퟐퟐퟖ ퟎ

ퟎ ퟓ.ퟑퟕퟐퟐퟕ

[ B ] = [ B .] = −ퟎ.ퟏퟓퟐퟎퟔ ퟎퟎ ퟏퟑ.ퟏퟓퟐퟎퟔ

Calculated by the use of online calculator

For Rectangular Matrices Let A = [ 2 4] and B = ퟑ ퟏ

ퟓ ퟔ are (1x2) and (2x2) matrices respectively. [ A ] [ B ] = [ C ]= [ 26 26 ]

ii) Assuming Matrix-A is unknown ( am x an ) = [ C ] / [ B] = ( 1 x 2) / ( 2 x 2) Columns are equal i.e., am x an = Dividend row X Divisor Row = ( 1 x 2 ) Form a matrix-X with order (1x2) [ X1 X2 ]

ퟑ ퟏퟓ ퟔ =[ 26 26 ]


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Linear Equations are 3X1 + 5X2 = 26 1X1 + 6X2 = 26 [ X1 X2 ] = [ 2 4 ]

i) Assuming Matrix-B is unknown ( bm x bn ) = [ C ] / [ A ] = (1 x 2) / ( 1 x 2) Row and Columns are Equal

Rows are equal bm x bn = Divisor column X Dividend column = ( 2 x 2 ) [ 2 4 ] 풙ퟏ 풙ퟐ

풙ퟑ 풙ퟒ =[ 26 26 ] 2.x1 + 4.x3 = 26 2x2 + 4.x4 = 26 x1 = ( 26 – 4 x3) / 2 & x2 = ( 26 – 4 x4) / 2

Columns are equal i.e., bm x bn = Dividend row X Divisor Row = (1x1) This is not correct because 2X=26 and 4X=26 X value cannot be same for both equations. Final answer is ( bm x bn) = (2x2) 5. IMPORTANT NOTES Note 1: Wrong Matrix Division gives wrong Quotient

(this is possible only in case of Square Matrices).

Note 2: This method is applicable for Rectangular matrices too if Determinant of Rectangular matrix is calculatable. Note 3: In both methods and both cases, Wrong Assumptions of Rows and Columns give Wrong Quotient Matrix which is Mirror Image Matrix clearly Note 4: Accurate matrices division is possible in Rectangular matrices only. Note 5: It is possible to create matrices like AB=BC where A≠C

Note 7: for Commutative Matrices (The matrices which satisfies Commutative Law) Division is possible by Methods and Both assumptions (i.e., Rows are Equal and columns are Equal).

Commutative Matrices: [ A ] [ B ] = [ B ] [ A] = [ C ] Where matrix – B equals to

i) [ B ] = [ A ] ii) [ B ] = [ A-1 ] iii) [ B ] = [ I ] iv) [ B ]=[ S ],Where [ S ] is a Diagonal

Matrix v) [ B ] = [ X ],Where [ X ] is Commutative

matrix to matrix [ A ]

i, ii, iii and iv always satisfies Commutative law [ A ] [ B ] = [ B ] [ A ]

For iv condition [ A ] [ B ] = [ B ] [A ] where [ B ] = [ X ], [ X ] is Commutative matrix to matrix [ A ]

6 CONCLUSION

In Matrices Division determinant Divisor must not be zero to divide two matrices. Division of Square matrices gives Mirror Image Matrix Which should obey the all the properties like Determinant, Trace, Law of Matrix Multiplication and Digonalization of Mirror image Matrix. Mirror Image matrix is also called as Pseudo or Slash Matrix. The main aim of these methods is to prove everything is possible in Mathematics and Physics with Equations to analyze the system. This Methods are based on Cramer’s Rule ( Substitution of column and also Rows ) and Gauss – Seidel Method ( to solve linear equations ). The Accuracy is 100% in Both cases but generates Mirror Image Matrices User has to select which one is suitable. The Time taken to calculate Real and Mirror Image Matrices is More and Memory Required to Calculate Both Matrices is More. For Commutative matrices both real and mirror image matrices are same.


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7 REFERENCES

[ 1 ] Bachmann, F.; Schmidt, E.: n-Ecke. B. I. Hochschultaschenbuch 471/471a, Mannheim, Wien, Zürich 1970. Zbl 0208.23901 [ 2 ] Radić, M.: A Definition of Determinant of Rectangular Matrix. Glas. Mat. 1(21) (1966), 17 – 22. Zbl 0168.02703 [ 3 ] Suˇsanj, R.; Radić, M.: Geometrical Meaning of One Generealization of the Determinant of Square Matrix. Glas. Mat., III.Ser. 29(2) (1994), 217 – 233. Zbl 0828.15005

Math. 1356, Springer - Verlag, 1988

Math. Monthly 62 (1955) 395-413.

[ 4 ] Yaglom, I. M.: Complex Numbers in Geometry. Translated by E. J. F. Primrose, Aca-demic Press, New York 1966. Zbl 0147.20201 [ 5 ] Neelam Jeevan Kumar, “matrices division” india, vol - 1, pp. 570-578, sept. 2012, ISSN: 2278 - 8697 [6] W.B Jurkat and H.G Ryser, “Matrix factorizasation and Determinants of permanents”, J algebra 3,1966,1-27 [ 7 ] Alston S. householdr, “The theory of matrices in numerical analysis” (blasidel gin and company) Newyork, Toronto ,London ,1964 [ 8 ] G. Boutry, M. Elad, G. Golub, P. Milanfar, The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach, SIAM J. Matrix Anal. Appl. 27 (2005), 582–601 [ 9 ] D. Chu, G. Golub, On a generalized eigenvalue problem for nonsquare pencils. SIAM J. Matrix Anal. Appl. 28 (2006), 770 – 787 [ 10 ] H. Volkmer, Multiparameter eigenvalue problems and expansion theorems. Lect. Notes.

[ 11 ] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997 [ 12 ] G. H. Golub and C. F. Van Loan, Matrix Computations, Third Edition, The Johns Hopkins University Press, Baltimore, 1996 [ 13 ] G. W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973 [ 14 ] G. W. Stewart, Matrix Algorithms: Basic Decompositions, SIAM, Philadel - phia, 1998 [ 15 ] L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997 [ 16 ] G Forsythe and C.B Moler “Computational System and Linear Algebraic Systems” Prentice - hall [ 17 ] J.H. Wilkinson, The Algebraic Eigen value Problem, Oxford University Press, Oxford, 1965 [ 18 ] Joshi V.A., A determinant For Rectangular Matrices, Bull. Austual. Math society., 21, 1980, 107 – 122 [ 19 ] Radic.M “A definition of Rectangular Matrix”, Glasnik, Matematicki, 1 ( 21 ), no. 1, 1966, 17 – 22. [ 20 ] Rinehart, R.F ., The Equivalence of Definitions of a Matrix Function, American


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division of matrices and mirror image ......rows of matrices j,k and s are equal to the columns of...

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