1 資訊科學數學 14 : determinants & inverses 陳光琦助理教授 (kuang-chi chen)...
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資訊科學數學資訊科學數學 14 14 ::
Determinants & InversesDeterminants & Inverses
陳光琦助理教授 陳光琦助理教授 (Kuang-Chi Chen)(Kuang-Chi Chen)[email protected]@mail.tcu.edu.tw
22
Linear Equations and MatricesLinear Equations and Matrices
DeterminantsDeterminants
33
3.1 Determinants3.1 Determinants• With each With each nnnn matrix matrix AA it is possible to associate a it is possible to associate a
scalar scalar det(det(AA)), called the , called the determinantdeterminant of the matrix, w of the matrix, whose value will tell us whether the matrix is hose value will tell us whether the matrix is singular singular or notor not..
• Case 1: 1Case 1: 11 matrices 1 matrices
- If - If AA = ( = (aa), then ), then AA will have a multiplicative inverse will have a multiplicative inverse iff iff aa≠0 ≠0 ..
- - AA is nonsingular iff det( is nonsingular iff det(AA))≠≠0 .0 .
44
222 Matrices2 Matrices
• Case 2: 2Case 2: 22 matrices 2 matrices
- Let - Let AA = . = .
- - AA will be will be nonsingularnonsingular iff iff det(det(AA) = ) = aa1111aa2222 – – aa1212aa2121≠ ≠ 00 . .
11 12
21 22
a a
a a
55
333 Matrices3 Matrices• Case 3: 3Case 3: 33 matrices 3 matrices
- Let - Let AA = . = .
- - AA will be will be nonsingularnonsingular iff iff
det(det(AA) = ) = aa1111aa2222aa3333 + + aa1212aa3131aa23 23 + + aa1313aa2121aa3232 – – aa1111aa3232aa23 23 – – aa1212
aa2121aa33 33 – – aa1313aa3131aa22 22 ≠ ≠ 00 . .
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
66
Example 4 & 5Example 4 & 5
• Example 4Example 4
If If AA = [ = [aa1111] is a 1] is a 11 matrix, then det(1 matrix, then det(AA) = ) = aa1111 . .
• Example 5Example 5
IfIf
⇒⇒ det(det(AA) = ) = aa1111aa2222 – – aa1212aa2121
⇒⇒ det(det(AA) = (2)(5) – (-3)(4)) = (2)(5) – (-3)(4) = 22 = 22
11 12
21 22
a aA
a a
54
32A
77
Example 6 & 7Example 6 & 7• Example 6Example 6
IfIf
⇒⇒ det(det(AA) =) = aa1111aa2222aa3333 + + aa1212aa3131aa23 23 + + aa1313aa2121aa3232
– – aa1111aa3232aa23 23 – – aa1212aa2121aa33 33 – – aa1313aa3131aa22 22
• Example 7Example 7
IfIf ⇒ ⇒ det(det(AA) = (1)(1)(2) + (3)(2)(1)) = (1)(1)(2) + (3)(2)(1) + (2)(3)(3)+ (2)(3)(3)
– – (3)(1)(3)(3)(1)(3) – (1)(1)(3)– (1)(1)(3) – (2)(2)(2) = 6– (2)(2)(2) = 6
333231
232221
131211
aaa
aaa
aaa
A
213
312
321
A
88
Properties of DeterminantsProperties of Determinants
• Theorem 3.1Theorem 3.1
The determinants of a matrix and its The determinants of a matrix and its transposetranspose are are eqequalual, i.e., det(, i.e., det(AA) = det() = det(AATT).).
99
Example 8Example 8• Example 8Example 8
IfIf
⇒ ⇒ det(det(AATT) = (1)(1)(2) + (3)(1)(2)) = (1)(1)(2) + (3)(1)(2) + (2)(3)(3)+ (2)(3)(3)
– – (3)(1)(3)(3)(1)(3) – (1)(1)(3)– (1)(1)(3) – (2)(2)(2)– (2)(2)(2)
= 6 = det(= 6 = det(AA))
233
112
321TA
213
312
321
A
1010
Theorem 3.2 & 3.3Theorem 3.2 & 3.3
• Theorem 3.2Theorem 3.2
If matrix If matrix BB results from matrix results from matrix AA by by interchanginginterchanging t two rows (or two columns) of wo rows (or two columns) of AA, then, then
det(det(BB) = -det() = -det(AA).).
• Theorem 3.3Theorem 3.3
If If two rows (or columns) of two rows (or columns) of AA are equal are equal, then, then
det(det(AA) = 0.) = 0.
1111
Example 9 & 10Example 9 & 10• Example 9Example 9
IfIf
• Example 10Example 10
IfIf
712
23 and 7
23
12
0
321
701
321
1212
Theorem 3.4Theorem 3.4
• Theorem 3.4Theorem 3.4
If a row (or column) of If a row (or column) of AA consists entirely of consists entirely of zeroszeros, t, then det(hen det(AA) = 0.) = 0.
• Example 11Example 11
0
000
654
321
1313
Theorem 3.5Theorem 3.5• Theorem 3.5Theorem 3.5
If If BB is obtained from is obtained from AA by by multiplyingmultiplying a row (colum a row (column) of n) of AA by a real number by a real number cc, then, then
det(det(BB) = ) = cc det( det(AA)) . .
• Example 12Example 12
1814641
1132
121
312
121
62
1414
Example 13Example 13
• Example 13Example 13
0032
141
151
121
32
341
351
321
2
682
351
321
1515
Theorem 3.6Theorem 3.6• Theorem 3.6Theorem 3.6
If If BB = [ = [bbijij] is obtained from ] is obtained from AA = [ = [aaijij] by adding to eac] by adding to eac
h element of the h element of the rrthth row (column) of row (column) of AA a constant a constant cc ti times the corresponding element of the mes the corresponding element of the ssthth row (colum row (column) n) rr≠≠ss of of AA, then det(, then det(BB) = det() = det(AA) .) .
• Example 14Example 14
101
312
905
101
312
321
1616
Theorem 3.7Theorem 3.7• Theorem 3.7Theorem 3.7
If a matrix If a matrix AA = [ = [aaijij] is ] is upper (lower) triangularupper (lower) triangular, then, , then,
then det(then det(AA) = ) = aa11 11 aa22 22 … … aannnn . .
• Corollary 1.3Corollary 1.3
The determinant of a The determinant of a diagonal matrixdiagonal matrix is the product o is the product of the entries on its main diagonal.f the entries on its main diagonal.
1717
Example 15Example 15
• Example 15Example 15
2 3 4 3 0 0 5 0 0
0 4 5 , 2 5 0 , 0 4 0
0 0 3 6 8 4 0 0 6
A B C
120)det( ,60)det( ,24)det( CBA
1818
Elementary OperationsElementary Operations• Elementary row and elementary column operationsElementary row and elementary column operations
I - Interchange rows (columns) I - Interchange rows (columns) ii and and jj : :
rrii ⇔ ⇔ rrjj ( (ccii ⇔ ⇔ ccjj ) )
II - Replace row (column) II - Replace row (column) ii by a nonzero value by a nonzero value kk times times row (column) row (column) ii : :
krkrii ⇔ ⇔ rrii ( (kckcii ⇔ ⇔ ccii ) )
III - Replace row (column) III - Replace row (column) jj by a nonzero value by a nonzero value kk times times row (column) row (column) i+ i+ row (column) row (column) jj : :
krkrii + + rrjj ⇔ ⇔ rrjj ( (kckcii + + ccjj ⇔ ⇔ ccjj ) )
1919
… … then …then …
det( ) det( ),
det( ) det( )
det ( ) det( ),
i j
i i
i j j
r r
kr r
kr r r
A A i j
A k A
A A i j
2020
Example 16Example 16• E.g. 16E.g. 16
642
523
234
A
13 32
1 3
4 3 2
det( ) 2det( ) 2det ( 3 2 5 )
1 2 3
4 3 2 1 2 3
2det( 3 2 5 ) ( 1) 2det( 3 2 5 )
1 2 3 4 3 2
r r
r r
A A
2121
Example 16Example 16 (cont’d)(cont’d)
1 2 2
1 3 3
52 3 38
-3r4
304
1 2 3 1 2 3
2det( 3 2 5 ) 2det( 0 -8 4 )
4 3 2 0 -5 10
1 2 3 1 2 3
2det( 0 -8 4 ) 2det( 0 -8 4 )
0 -5 10 0 0
r rr r r
r r r
det( ) 2(1)( 8)( 30 / 4) 120A
2222
Theorem 3.8Theorem 3.8• Theorem 3.8Theorem 3.8
The determinant of a product of two matrices is the prThe determinant of a product of two matrices is the product of their determinants oduct of their determinants det(det(ABAB) = det() = det(AA)det()det(BB)) . .
• Example 17Example 17
43
21A
21
12B
2A 5B
510
34AB BAAB 10
2323
Example 17 Example 17 (cont’d)(cont’d)
• RemarkRemark
ABAB≠≠BABA
||BABA| = || = |BB| || |AA|= -10 = ||= -10 = |ABAB||
107
01BA
2424
Corollary 3.2Corollary 3.2
• Corollary 3.2Corollary 3.2
If If AA is is nonsingularnonsingular, then , then det(det(AA) ) ≠≠ 0 0,,
thus thus det(det(AA-1-1) = 1/det() = 1/det(AA))..
If If AA is singular, then det( is singular, then det(AA) = 0) = 0
( 1 = |( 1 = |II| = || = |AAAA-1-1| = || = |AA| || |AA-1-1| )| )
2525
Example 18Example 18• Example 18Example 18
43
21A 2)det( A
1 2 1
3/ 2 1/ 2A
)det(
1
2
1)det( 1
AA
2626
Cofactor Expression and Cofactor Expression and ApplicationsApplications
2727
3.2 Cofactor Expression and 3.2 Cofactor Expression and ApplicationsApplications
Cofactor expression and applicationsCofactor expression and applications
• Definition – Minor and cofactorDefinition – Minor and cofactor
Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. Let matrix. Let MMijij be the ( be the (nn-1)-1)
((nn-1) -1) submatrixsubmatrix of of AA obtained by deleting the obtained by deleting the iithth row row and and jjthth column of column of AA. The . The determinant det(determinant det(MMijij)) is calle is calle
d the d the minorminor of of aaijij. The . The cofactor cofactor AAijij of of aaijij is defined as is defined as
)( det)1( ijji
ij MA
2828
Example 1Example 1• E.g. 1E.g. 1
LetLet
217
654
213
A
12
23
31
4 6det( ) 8 42 34
7 2
3 1det( ) 3 7 10
7 1
1 2det( ) 6 10 16
5 6
M
M
M
1 212 12
2 323 23
3 131 31
( 1) det( ) ( 1)( 34) 34
( 1) det( ) ( 1)(10) 10
( 1) det( ) (1)( 16) 16
A M
A M
A M
2929
Theorem 3.9Theorem 3.9
• Theorem 3.9Theorem 3.9
Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. Then matrix. Then
for each 1for each 1≤ ≤ ii ≤ ≤ nn,,
det(det(AA) = ) = aaii11AAii11 + + aaii22AAii22 + … + + … + aaininAAinin , and , and
for each 1for each 1≤ ≤ jj ≤ ≤ nn,,
det(det(AA) = ) = aa11jjAA11jj + + aa22jjAA22jj + … + + … + aanjnjAAnjnj . .
3030
Example 2Example 2To evaluate the determinantTo evaluate the determinant
3202
3003
3124
4321
3 1 3 2
3 3 3 4
1 2 3 42 3 4 1 3 4
4 2 1 3( 1) (3) 2 1 3 ( 1) (0) 4 1 3
3 0 0 30 2 3 2 2 3
2 0 2 3
1 2 4 1 2 3
( 1) (0) -4 2 3 ( 1) ( 3) 4 2 1
2 0 3 2 0 2
( 1)(3)(20) 0 0 ( 1)( 3)( 4) 48
3131
Example 3Example 3Consider the determinant of the matrixConsider the determinant of the matrix
4 1 4
1 2 1
3 1 4
4
1 2 3 4 1 2 3 5
4 2 1 3 4 2 1 1
3 0 0 3 3 0 0 0
2 0 2 3 2 0 2 5
2 3 5 0 4 6
( 1) (3) 2 1 1 ( 1) (3) 2 1 1
0 2 5 0 2 5
( 1) (3)( 2)( 8) 48
c c c
r r r
3232
Theorem 3.10Theorem 3.10• Theorem 3.10Theorem 3.10
If If AA = [ = [aaijij] be an ] be an nnnn matrix, then matrix, then
aaii11AAkk11 + + aaii22AAkk22 + … + + … + aaininAAknkn = 0, for = 0, for ii≠≠kk , ,
aa11jjAA11kk + + aa22jjAA22kk + … + + … + aanjnjAAnknk = 0, for = 0, for jj≠≠kk . .
3333
Example 4Example 4• E.g. 4E.g. 4
033142191
032145194
354
211
1424
311
1925
321
254
132
321
231322122111
233322322131
3223
2222
1221
AaAaAa
AaAaAa
A
A
A
A
3434
AdjointAdjoint• Definition – AdjointDefinition – Adjoint
Let Let AA = [ = [aaijij] be an ] be an nnnn matrix. The matrix. The nnnn matrix matrix adjadj AA,,
called the called the adjoint of adjoint of AA, is the matrix whose , is the matrix whose jj, , iithth ele element is the ment is the cofactor cofactor AAijij of of aaijij . Thus . Thus
nnnn
n
n
AAA
AAA
AAA
adjA
21
22212
12111
3535
RemarkRemark
• RemarkRemark
The adjoint of The adjoint of AA is formed by taking the is formed by taking the transptransposeose of the matrix of of the matrix of cofactorscofactors AAijij of the elemen of the elemen
ts of ts of AA..
3636
Example 5Example 5• Example 5Example 5
Compute adj Compute adj AA
301
265
123
A
3737
SolutionSolution
1 1
11
1 2
12
1 3
13
2 1
21
2 2
22
2 3
23
6 21 18
0 3
5 21 17
1 3
5 61 6
1 0
2 11 6
0 3
3 11 10
1 3
3 21 2
1 0
A
A
A
A
A
A
2865
231
125
131
1026
121
3333
2332
1331
A
A
A
2826
11017
10618
adj Then, A
3838
Theorem 3.11Theorem 3.11
• Theorem 3.11Theorem 3.11
If If AA = [ = [aaijij] be an ] be an nnnn matrix, then matrix, then
AA(adj (adj AA) = (adj ) = (adj AA))AA = det( = det(AA) ) IInn . .
3939
Example 6Example 6• E.g. 6 E.g. 6
Consider the matrixConsider the matrix
3 2 1 18 6 10 94 0 0 1 0 0
5 6 2 17 10 1 0 94 0 94 0 1 0
1 0 3 6 2 28 0 0 94 0 0 1
and
18 6 10 3 2 1 1 0 0
17 10 1 5 6 2 94 0 1 0
6 2 28 1 0 3 0 0 1
301
265
123
A
4040
Corollary 3.3Corollary 3.3• Corollary 3.3Corollary 3.3
If If AA = [ = [aaijij] be an ] be an nnnn matrix and det( matrix and det(AA))≠≠0, then0, then
A
A
A
A
A
A
A
A
A
A
A
AA
A
A
A
A
A
adjAA
A
detdetdet
detdetdet
detdetdet
det
1 1
4141
Example 7Example 7• Example 7Example 7
Consider the matrixConsider the matrix
Then det(Then det(AA) = -94, and) = -94, and
3 2 1
5 6 2
1 0 3
A
18 6 1094 94 94
1 17 10 194 94 94
6 128294 94 94
1
det( )A adjA
A
4242
Theorem 3.12Theorem 3.12
• Theorem 3.12Theorem 3.12
A matrix A matrix AA = [ = [aaijij] is ] is nonsingularnonsingular iff iff det(det(AA) ) ≠≠ 0 0..
• Corollary 3.4Corollary 3.4
For an For an nnnn matrix matrix AA, the homogeneous system , the homogeneous system AAx x = 0 has a = 0 has a nontrival solutionnontrival solution iff iff det(det(AA) = 0) = 0..
4343
Example 8Example 8• Example 8Example 8
Let Let AA be a 4x4 matrix with det( be a 4x4 matrix with det(AA) = -2) = -2
(a) describe the set of all solutions to the homoge(a) describe the set of all solutions to the homogeneous system neous system AAxx = 0. = 0.
(b) If (b) If AA is transformed to reduced row echelon form is transformed to reduced row echelon form BB, , what is what is BB??
(c) Given an expression for a solution to the linear syst(c) Given an expression for a solution to the linear system em AAxx = = bb, where , where bb = [ = [bb11 , , bb22 , , bb33 , , bb44 ] ]TT . .
(d) Can the linear system (d) Can the linear system AAxx = = bb have more than one sol have more than one solution? Explain.ution? Explain.
(e) Does (e) Does AA-1-1 exist? exist?
4444
Solutions of Example 8Solutions of Example 8• SolutionsSolutions
(a) Since det((a) Since det(AA))≠≠0, 0, AxAx = 0 has = 0 has only the trivial solutiononly the trivial solution..
(b) Since det((b) Since det(AA))≠≠0, 0, AA is a nonsingular matrix, so is a nonsingular matrix, so BB = = IInn
(c) A solution to the given system is given by (c) A solution to the given system is given by xx = = AA-1-1bb
(d) No. The solution is unique.(d) No. The solution is unique.
(e) Yes. (e) Yes.
4545
Nonsingular EquivalenceNonsingular Equivalence• List of nonsingular equivalenceList of nonsingular equivalence
The following statements are equivalent.The following statements are equivalent.
1.1. AA is is nonsingularnonsingular..
2.2. xx = 0 = 0 is the is the only solutiononly solution to to AxAx = 0. = 0.
3.3. AA is is row equivalencerow equivalence to to IInn . .
4. The linear system 4. The linear system AAxx = = bb has a has a unique solutionunique solution for for every every nn1 matrix 1 matrix bb..
5. 5. det(det(AA))≠≠00 . .
4646
DeterminantsDeterminants
• Linearly independentLinearly independent
• NonsingularNonsingular
• Trivial solution Trivial solution xx = 0 to = 0 to AxAx = 0 = 0
• det(det(AA) ) ≠≠ 0 0
4747
DeterminantsDeterminants
• Linearly dependentLinearly dependent
• SingularSingular
• Nontrivial solution to Nontrivial solution to AAxx = 0 = 0
• det(det(AA) = 0) = 0
4848
Cramer’s RuleCramer’s RuleTheorem 3.13 (Cramer’s Rule)Theorem 3.13 (Cramer’s Rule)
Let Let aa1111xx11 + + aa1212xx22 + … + + … + aa11nnxxnn = = bb11
aa2121xx11 + + aa2222xx22 + … + + … + aa22nnxxnn = = bb22
……
aann11xx11 + + aann22xx22 + … + + … + aannnnxxnn = = bbnn
Then,Then,
xx11 = det( = det(AA11)/det()/det(AA)) , , xx22 = det( = det(AA22)/det()/det(AA)) , … , , … ,
xxnn = det( = det(AAnn)/det()/det(AA)) . .
4949
Cramer’s RuleCramer’s Rule
Cramer’s Rule for solving the linear system Cramer’s Rule for solving the linear system AAxx = = bb, wh, where ere AA is is nnnn, is as follows:, is as follows:
Step 1. Compute det(Step 1. Compute det(AA). If det(). If det(AA) = 0, Cramer’s rule is ) = 0, Cramer’s rule is not applicable. Use not applicable. Use Gauss-Jordan ReductionGauss-Jordan Reduction..
Step 2. If det(Step 2. If det(AA))≠≠0, for each 0, for each ii,,
xxii = det( = det(AAii)/det()/det(AA)) , ,
where where AAii is the matrix obtained from is the matrix obtained from AA by replacing t by replacing t
he he iithth column of column of AA by by bb. .
5050
Example 9Example 9
• Consider the following linear system:Consider the following linear system:
-2-2xx11 + 3 + 3xx22 – – xx33 = 1 = 1
xx11 + 2 + 2xx22 – – xx33 = 4 = 4
-2-2xx11 – 2 – 2xx22 + + xx33 = -3 = -3
ThenThen 2 3 1
1 2 1 2
2 1 1
A
5151
Example 9Example 9 (cont’d) (cont’d)
2
2 1 1
1 4 1
2 3 1 63
2x
A
1
1 3 1
4 2 1
3 1 1 42
2x
A
3
2 3 1
1 2 4
2 1 3 84
2x
A
Hence,Hence,
5252
Polynomial Interpolation Polynomial Interpolation RevisitedRevisited
• Polynomial Interpolation RevisitedPolynomial Interpolation Revisited
To find a quadratic polynomial that interpolates the To find a quadratic polynomial that interpolates the following points:following points:
((xx11, , yy11), (), (xx22, , yy22), (), (xx33, , yy33),),
where where xx11≠≠xx22 , , xx11≠ ≠ xx33 , , xx22≠ ≠ xx33 . .
5353
… … more …more …
The polynomial has the form:The polynomial has the form:
yy = = aa22xx22 + + aa11xx + + aa00 . .
The corresponding linear systemThe corresponding linear system
yy11 = = aa22xx1122 + + aa11xx11 + + aa00 , ,
yy22 = = aa22xx2222 + + aa11xx22 + + aa00 , ,
yy33 = = aa22xx3322 + + aa11xx33 + + aa00 . .
5454
… … more …more …
The coefficient matrixThe coefficient matrix
The Vandermount determinantThe Vandermount determinant
((xx11 – – xx22 )( )( xx11 – – xx33 )( )( xx22 – – xx33 ) )
1
1
1
3
2
3
2
2
2
1
2
1
xx
xx
xx
5555
5656
Linear Equations and MatricesLinear Equations and Matrices
LU-FactorizationLU-Factorization
5757
LU-FactorizationLU-Factorization
• 1.8 LU-Factorization1.8 LU-Factorization
If a square matrix can be reduced to upper If a square matrix can be reduced to upper triangular form using only 3 row operations, triangular form using only 3 row operations, then it is possible to represent the then it is possible to represent the reduction reduction processprocess in terms of a matrix factorization. in terms of a matrix factorization.
5858
Type I OperationType I Operation
An elementary matrix of type I is a matrix obtained by An elementary matrix of type I is a matrix obtained by interchanginginterchanging two rows of identity matrix two rows of identity matrix II..
ExampleExample1
1 1
1
1
0 1 0 0 1 0
1 0 0 , 1 0 0
0 0 1 0 0 1
Interchange the first row and
the second row of
Interchange the first column
and the second column of
E E
E A
A
AE
A
5959
Type II OperationType II Operation
An elementary matrix of type II is a matrix obtained An elementary matrix of type II is a matrix obtained by by multiplyingmultiplying a row of a row of II byby a nonzero constant.a nonzero constant.
ExampleExample
12 2
2
2
1 0 0 1 0 0
0 1 0 , 0 1 0
0 0 3 0 0 1/ 3
Multiply the third row of by 3
Multiply the third column of by 3
E E
E A A
AE A
6060
Type III OperationType III Operation
An elementary matrix of type III is a matrix obtained An elementary matrix of type III is a matrix obtained from from II by adding a multiple of one row to another row. by adding a multiple of one row to another row.
ExampleExample1
3 3
3
3
1 0 3 1 0 3
0 1 0 , 0 1 0
0 0 1 0 0 1
Add three times the third row to
the first row of
Add three times the third column
to the first column of
E E
E A
A
AE
A
6161
In GeneralIn GeneralIn general, the elementary matrix by adding In general, the elementary matrix by adding mm times times of row of row ii to row to row jj
1000
10
10
1
,
1000
10
10
1
133
m
E
m
E
Row j
Column iaji
6262
TheoremTheorem• TheoremTheorem
If If AA and and BB are are nonsingular squarenonsingular square matrices, then matrices, then ABAB i is also s also nonsingularnonsingular..
i.e. (i.e. (ABAB))-1-1 = = BB-1-1AA-1-1 . .
In general, if In general, if EE11, , EE22, …, , …, EEkk are all are all nonsingularnonsingular, then t, then t
he product he product EE11EE22 … … EEkk is also is also nonsingularnonsingular and and
((EE11EE22 … … EEkk ) )-1-1 = = EEkk-1-1 … … EE22
-1 -1 EE11-1-1 . .
6363
Row EquivalenceRow Equivalence
• A matrix A matrix BB is is row equivalent torow equivalent to AA if there exists a fin if there exists a finite sequence ite sequence EE11EE22 … … EEkk of elementary matrices such of elementary matrices such
that that BB = = EEkk … … EE22 EE11AA . .
6464
LU-FactorizationLU-Factorization
• LU-FactorizationLU-Factorization
If a square matrix can be reduced to upper If a square matrix can be reduced to upper triangular form using only 3 row operations, triangular form using only 3 row operations, then it is possible to represent the then it is possible to represent the reduction reduction processprocess in terms of a matrix factorization. in terms of a matrix factorization.
6565
ExampleExample• LetLet
914
251
242
A
1
1 21
2 4 2 2 4 2
1 5 2 0 3 1
4 1 9 4 1 9
1 0 0
1/ 2 1 0 , 1/ 2
0 0 1
E A
E a
6666
(cont’d)(cont’d)
2
2 31
2 4 2 2 4 2
0 3 1 0 3 1
4 1 9 0 9 5
1 0 0
0 1 0 , 2
2 0 1
E A
E a
3
3 32
2 4 2 2 4 2
0 3 1 0 3 1
0 9 5 0 0 8
1 0 0
0 1 0 , 3
0 3 1
E A
E a
6767
(cont’d)(cont’d)
3 2 1
1 1 1 13 2 1 1 2 3
1 1 11 2 3
2 4 2
0 3 1
0 0 8
( )
1 0 01 0 0 1 0 0
11 0 0 1 0 0 1 0
22 0 1 0 3 10 0 1
E E E A U
A E E E U E E E U LU
L E E E