International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013 ISSN 2278-7763
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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
IJOART ⎡ ⎢ =⎢ : ⎢ : ⎣
…. ….
⎤ ⎥ ∶ ∶ ⎥ ∶ ∶ ⎥ …. ⎦
⎡ ⎢ = ⎢ ⎢ ⎣
…. ….
: :
∶ ∶
….
jmxjn
Where
∶ ∶
⎤ ⎥ ⎥ ⎥ ⎦
kmxkn
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 ) …. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥ ⎥ ⎦
General Condition to divide two matrices 1. 2.
Determinant of Divisor Should not be Zero (i.e., |A|/|B| = element-c. |B| ≠ 0 ) Either divisor row/column must be equal to dividend row/column
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Proof: for condition -2 Multiplicand is unknown i.e., [J] is unknown.
Multiplier is unknown i.e., [ K ] is unknown.
To find order matrix –J = [ J ]
To find order matrix –K =[ K ]
[J]=[S]/[K]
[ K ] = [ S ]/[ J ]
JM X JN = (SM X SN) / (KM X KN) = (JM X KN) / (KM X
KM X KN= (SM X SN) / (JM X JN) = (JM X KN) / (JM X
KN)
JN)
In this case Dividend and Divisor columns must be
In this case Dividend and Divisor rows must be Equal
Equal to get Quotient (i.e.,[ J ]).
to get Quotient(i.e.,[ K ]).
The order of Quotient matrix is
The order of Quotient matrix is
dividend row x divisor row
2. Methods i) ii)
divisor column x dividend column
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Determinants 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 [ K ]
the elements of [ J ] ,
→
[ ] , → = −−−−−−− [ ] ,
2.i.b when Columns of Dividend & Divisor are Equal
…
(
)
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 )
→
,
[ ] → , = −−−−−−− [ ] ,
)
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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….(
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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 ⎡ ⎢ ⎢ ⎢ ⎣
…. …. : :
∶ ∶
⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣
∶ ∶ ….
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎡ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣
…. …. : :
∶ ∶
∶ ∶ ….
2.ii.b when Columns of Dividend & Divisor are Equal ⎤ ⎥ ⎥ ⎥ ⎦
[ K ] and [ S ] are known matrices. [ X ] is unknown
⎡ ⎢ ⎢ ⎢ ⎣
…. ….
: :
⎤⎡ ⎥⎢ ∶ ∶ ⎥⎢ ∶ ∶ ⎥⎢ …. ⎦⎣
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥= ⎥ ⎦
⎡ ⎢ ⎢ ⎢ ⎣
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥ ⎥ ⎦
[ J ] and [ S ] are known matrices. [ X ] is unknown
matrix
matrix. Now solve the unknown elements of unknown matrix.
Now solve the unknown elements of unknown matrix. j11x11 + j12x21 + j13x31 +…….+ j1nxm1 = s11
IJOART
x11 k11 + x12k21 + x13 k31 +…………+ x1nkm1 = s11
j21x11 + j22x21 + j23x31 +…….+ j2nxm1 = s21
x11 k12 + x12k22 + x13 k32 +…………+ x1nkm2 = s12
j31x11 + j32x21 + j33x31 +…….+ j3nxm1 = s31
x11 k13 + x12k23+ x13k33 +…………+ x1nkm3 = s13
to
to
jm1x11 + jm2x21 + jm3x31 +…….+ jmnxm1 = sm1
x11 k1n + x12 k2n + x13 k3n +…………+ x1nkmn = s1n
Find x11 to x1n values
Find x11 to x1n values. And repeat same procedure to find remaining And repeat same procedure to find remaining
x21 to x2n values x21 to x2n values
x31 to x3n values
x31 to x3n values
to
to
xm1 to xmn values
xm1 to xmn values final matrix substitute the values in matrix-X which is equal to
substitute the values in matrix-X which is equal to
Matrix-J
Matrix-K
⎡ ⎢ ⎢ ⎢ ⎣
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎡ ⎥ ⎢ ⎥=⎢ : ⎥ ⎢ : ⎦ ⎣
…. ….
⎤ ⎥ ∶ ∶ ⎥ ∶ ∶ ⎥ …. ⎦
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⎡ ⎢ ⎢ ⎢ ⎣
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎡ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣
…. …. : :
IJOART
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥ ⎥ ⎦
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434
3. Mirror Image Matrix[b] In mirrors Reflection image is opposite to Real image. Mirror shows Right as left and vice versa.
Mirror [ A ][ B] = [C]
[ C] = [B.][A] [ C] = [B][A.]
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 )
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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
∑
∑
=
…..
,
(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. ] = ⎢ : ⎢ : ⎣ .
. . ∶ ∶ .
…. . …. . ∶ ∶ …. .
⎤ ⎥ ⎥ and ⎥ ⎦
⎡ ⎢ [A]= ⎢ ⎢ ⎣
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥ ⎥ ⎦
A. x = λ x
…..
( vi .a )
Ay=λy
…...
( vi. b )
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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013 ISSN 2278-7763
435
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 ] | .−
3. iii
|= | −
|=
……..
( vi. c )
Trace
Let matrix - A. = [ A. ] is the Mirror Image Matrix of Matrix – A = [ A ] . ⎡ . ⎢ [ A. ] = ⎢ : ⎢ : ⎣ .
. . ∶ ∶ .
…. . …. . ∶ ∶ …. .
⎤ ⎥ ⎥ and ⎥ ⎦
⎡ ⎢ [A]= ⎢ ⎢ ⎣
…. …. : :
∶ ∶
⎤ ⎥ ⎥ ⎥ ⎦
∶ ∶ ….
Definition: Trace is the sum of Diagonal Elements of a Matrix Trace ( A. ) = ∑ Trace ( A ) = ∑
.
= a.11 + a.22 + a.33 + a.44 + ….. + a.mn
………..
( vii. a)
………..
( vii. b )
IJOART = a11 + a22 + a33 + a44 + ….. + amn
Matrices A. is Mirror image to A
( vii. a ) = ( vii. b )
Trace ( A ) = Trace ( A. )
∑
3.iv
.
=∑
Matrix Digonalization[d]
Let matrix - A. = [ A. ] is the Mirror Image Matrix of Matrix – A = [ A ] . ⎡ . ⎢ [ A. ] = ⎢ : ⎢ : ⎣ .
. . ∶ ∶ .
…. . …. . ∶ ∶ …. .
⎤ ⎥ ⎥ and ⎥ ⎦
⎡ ⎢ [A]= ⎢ ⎢ ⎣
P-1 A P = D1 Q-1 A. Q = D2
…. …. : :
∶ ∶
∶ ∶ ….
⎤ ⎥ ⎥ ⎥ ⎦
…………..
( viii .a )
…………..
( 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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International Journal of Advancements in Research & Technology, Volume 2, Issue 7, July-2013 ISSN 2278-7763
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436
Example
For Square Matrices Let [A] =
i)
[ B ] = [ C ] / [ A ] = ( 2 x 2 ) / ( 2 x 2)
are 2x2 matrices
[B]=
respectively.
Assuming matrix-B is unknown
ii.a) Rows are Equal (i.e., Km = Sm)
The product of [ A ] [B ] = [ C ] = →
Determinant of matrices A and B are -2 and -2 | i)
|=|
,
[ ] → , = −−−−−−− [ ] ,
| = −2
Assuming matrix-A is unknown [A]=[C ]/[B]=(2x2)/(2x2)
b11=
= -10/-2 = 5 : a12 =
b21 =
= -14/-2=7 : a22 =
= -88/-2 = 6
= -16/-2 = 8
i.a) Rows are Equal (i.e., Km = Sm) →
[ ] → , = −−−−−−− [ ] ,
,
= -106/-2 = 53 : a12 =
a21 =
= 82/-2= - 41 : a22 =
ii.b) Columns are Equal (i.e., Kn = Sn)
=
= -124/-2 = 62
→
=
,
→
=
[ ] , → −−−−−−− [ ] ,
= 96/-2 = -48
−
−
i.b) Columns are Equal (i.e., Kn = Sn)
,
=
IJOART
a11=
[ A.] =
[ B] =
a11=
= 10/-2 = -5 : a12=
a21=
= 22/-2 = -11 : a22=
[ ] , → −−−−−−− [ ] ,
[
]=
= -16/-2 = 8 = -36/-2 = 18
=
− −
[ A ][ B ] = [ B. ][ A] = [ C ] a11=
= -2/-2 = 1 : a12=
= -4/-2 = 2
a21=
= -6/-2 = 3 : a22=
= -8/-2 = 4
[
]=
=
[ A ] [ B ] = [ B ] [A. ] = [ C] [A] and [A.] are follows Mirror image matrix[b]
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[ B ] and [ B.] are follows Mirror image matrix[b] Law of Matrix Multiplication
[A]=
[B]= [ A ][ B ] = [ C ] =
(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
− −
− −
437
=0
2
- 13λ – 2 = 0
λ1 = 13.152067347825, λ2 = -0.152067347825 [ B. ][ A] = [ C ]
Eigen Vales of Matrix-B.
(-5-11 )( 2+1 )+( 18+8 )( 3+4 ) = ( 19+22+43+50
| λI − K. | = 0
)=134 +
Trace
− −
2
=0
- 13λ – 2 = 0
Tr( A ) = sum of Diagonal elements of Matrix - A
λ1 = 13.152067347825, λ2 = -0.152067347825 Tr( A ) = 1+4 = 5
Tr( A. ) = 53-48 = 5 From the Calculated Eigen Values
Tr ( A ) = Tr ( A. ) Eigen Values of [A] and [A.] are Equal and Eigen Tr ( B ) = 5 + 8 = 13
Tr ( B.) = -5 + 18 = 13
Values of [B] and [B.] are Equal.
Tr ( B ) = Tr ( B. ) Matrix Digonalization
From the Calculated Traces
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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
[ A ] = [ A. ] =
[ B ] = [ B .] =
− .
.
− .
.
Calculated by the use of online calculator
Eigen Vales of Matrix-J
| λI − A | = 0 − −
− −
=0
2
- 5λ – 2 = 0
λ1 = -0.3372281323269, λ2 = 5.372281323269
Eigen Value of Matrix- A. | λI − A. | = 0
−
− +
=0
2
- 5λ – 2 = 0
λ1 = -0.3372281323269, λ2 = 5.372281323269
Eigen Vales of Matrix-B
For Rectangular Matrices
Let A = [ 2 4] and B =
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 ]
=[ 26 26 ]
| λI − K | = 0
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Linear Equations are
3X1 + 5X2 = 26 1X1 + 6X2 = 26
438
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).
[ X1 X2 ] = [ 2 4 ] Commutative Matrices:
i)
Assuming Matrix-B is unknown [ A ] [ B ] = [ B ] [ A] = [ C ]
( bm x bn ) = [ C ] / [ A ] = (1 x 2) / ( 1 x 2) Row and Columns are Equal
Rows are equal
Where matrix – B equals to i) ii) iii) iv)
bm x bn = Divisor column X Dividend column = ( 2 x 2 v)
) [2 4]
=[ 26 26 ]
2.x1 + 4.x3 = 26 2x2 + 4.x4 = 26
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
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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.
[B]=[A] [ B ] = [ A-1 ] [B]=[I] [ B ]=[ S ],Where [ S ] is a Diagonal Matrix [ B ] = [ X ],Where [ X ] is Commutative matrix to matrix [ A ]
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
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.
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
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439
REFERENCES
[ 1 ] Bachmann, F.; Schmidt, E.: n-Ecke. B. I. Hochschultaschenbuch 471/471a, Mannheim, Wien, Z¨urich 1970. Zbl 0208.23901 [ 2 ] Radi´c, M.: A Definition of Determinant of Rectangular Matrix. Glas. Mat. 1(21) (1966), 17 – 22. Zbl 0168.02703 [ 3 ] Suˇsanj, R.; Radi´c, M.: Geometrical Meaning of One Generealization of the Determinant of Square Matrix. Glas. Mat., III.Ser. 29(2) (1994), 217 – 233. Zbl 0828.15005 [ 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. Math. 1356, Springer - Verlag, 1988 [ 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 Math. Monthly 62 (1955) 395-413.
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