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1
CHAPTER 1
Systems of Linear Equations and
Matrices
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prepared by Razana Alwee
2
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(values s1,s2,..snare substitute for x1,x2,.xn)
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Solution
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Example
Determine whether either of the points (1,5)
and (0,2) is a solution to the given system of
equations.
y= 3x2
y=x6
prepared by Razana Alwee
8
To check the given possible solutions, just plug the x-and y-coordinates into the equations, and check to
see if they work.
How?
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Solution
checking (1,5):
(5) = 3(1)2
5 =325 =5 (solution checks)
(5)=(1)6
5 = 16
5 =5 (solution checks)
Since the given point works in each equation, it is a
solution to the system.
prepared by Razana Alwee
9
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(unique or infinite solutions)
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Unique Solution
A system,
The point where the lines cross {x=3.6, y=0.4} is thesolution that satisfies both equations simultaneously.
The solution is unique and consistent.
13
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Consistent
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Infinite Many Solutions
A systems,
This system is redundant because the second equation
is equivalent to the first one.
They 'cross' at an infinite number of points, so there are
an infinite number of solutions
Consistent
prepared by Razana Alwee
15
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(Consistent)
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No Solutions
A system
consists of two parallel lines that never cross. Thus
there is no solution.
Inconsistent 17
(1)
(2)
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(Inconsistent)
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Solving Linear System
Matrices are helpful in rewriting a linear system in a very
simple form.
Matrix form, AX= B
prepared by Razana Alwee
19
mnmm
n
n
aaa
aaa
aaa
A
:
:::::
:
21
22221
11211
nx
x
x
X:
2
1
nb
b
b
B:
2
1
,
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A matrix is said to be row-echelon form if the following
conditions are satisfied:
Every row with all 0 entries is below every row with
nonzero entriesEach leading entry of a row is in a column to the right of the
leading entry of the row above it(leading entry-any nonzero
value)
(The second row starts with more zero than the first(left to
right)All entries in a column below a leading entry are zeros.
Row-echelon Matrices
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prepared by Razana Alwee
24
0 0 0 0 0 0
0 0 0 0 0 #
0 0 # * * *
# * * * * *#-leading entry(left most
non zero entry)*-any number/values including 0
Row-echelon Matrices
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prepared by Razana Alwee
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(0s below and
above each
leading 1) 0 0 0 0 0 00 0 0 0 0 10 0 1 * * 0# * 0 * * 0
Reduced Row-echelon Matrices
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or row-echelon form
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Elementary Row Operation
Elementary row operations can be used to transform thematrix into its row-echelon form. If we denote row 1 byR1, row 2 by R2, etc., and ais a scalar, the threeelementary row operations are as follows:
1) Swap two rows (equations), denoted R1R2 (this wouldswap rows 1 and 2)
2) Multiply a row by a nonzero scalar, denoted aR1 (thiswould multiply row 1 by k)
3) Add a multiple of a row to another row, denoted R1 +aR2 (this would add ktime row 2 to row 1, and replacethe previous row 1)
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Example
Given is a linear equation system,
Augmented form
124
423
72
221
321
321
xxx
xxx
xxx
1
47
241
123112
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Example (cont.)
Swap two rows
14
7
241123
112R
1 R
3
74
1
112123
241
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Multiply row 1 with a =3 = 3R1and add the multiple row
to another row, R2
122 3RRR
7)1(34
7)2(31
14)4(32
0)1(33
7
71
112
7140241
3 122 RRR
7
4
1
112
123
241
The notation means to take row 2 and subtract 3 times
row 1 from it to produce the new augmented matrix.
Example (cont.)
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Bi Bi
Bi Bi
Bi +Bi Bj
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Solution
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Solution (cont.)
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Solution (cont.)
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Solution (cont.)
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Solution (cont.)
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Solution (cont.)
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X=(3,2,1)T
Solution (cont.)
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Solution
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Solution (cont.)
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Solution (cont.)
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(solve leading variables using back-substitutions)
(assign nonleading variables as parameter)
Solution (cont.)
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50
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Introduction to Matrix
A rectangular arrangement of number is called a matrix
Matrices are usually denoted using upper case letter A,
B, C, K, P, etc
example
A =
2 0 -4
3 1 7 B =
5 2 8
3 4 3
3 9 6
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Introduction to Matrix
The size of matrix is depend on the
number of rows and columns
A matrix with mrows and ncolumns is
called an m-by-nmatrix (m x nmatrix).
Example:
A =
2 0 -4
3 1 7
rows
columns
2
3
2 x 3 matrix
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Introduction to Matrix
A matrix of size 1 x nis called a row
matrix:
A matrix of size mx 1 is called a
column matrix:
C = -1 5 2
B =1
4
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prepared by Razana Alwee
54
A matrix with equal numbers of rows and
columns is called a square matrix:
E =7 3
8 -2C =
4 12 6
7 0 4
11 3 -5
Introduction to Matrix
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Example
C =
2 0
3 1B =
5 2 8
7 4 33 9 6 -4 9
D =
6
2-4
E =4 -5
8 1F = 6 2 9
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Introduction to Matrix
The numbers in the matrix are called its entries
The entry in the ith rowand jth column of
matrixXis referred as the (i,j)-entry of thematrixXand denoted by a lower case letter
with two subscripts indices,xi,j
If Ais an m x nmatrix. The (i,j)-entry of Ais
denoted by ai,j
A =a1,1 a1,2 a1,3 a1,n..a2,1 a2,2 a2,3 a2,n..
: : : :am,1 am,2am,3 am,n..
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Introduction to Matrix
Example: Ais an 2 x 3 matrix
A =
2 0 -4
3 1 7
The (1,2) entry of A is 0; a1,2= 0The (2,3) entry of A is 7;a2,3= 7
B =
5 2 8
7 4 3
3 9 6
The (3,1) entry of B is ?;
b1,3= ?
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Exercise
What is the (3,4)-entry?
9006489
5030323
344569
15350
X=
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Matrices
The diagonal entries in an mxnmatrix A=[aij] are a11,a22 ,a33and they form the main diagonalof A.
mnmmm
n
n
ij
aaaa
aaaa
aaaa
aA
..
:::::
..
..
321
2232221
1131211
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Matrices
A diagonal matrix is a square matrix whose
nondiagonal entries are zero.
Example: nxnidentity matrixIn
1000
0100
0010
0001
nI
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Matrices
The mxnmatrix whose entries are all zero is called the
zero matrixand will be denoted by 0.
0000
0000
0000
0000
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Matrices
Two matrices A and B are equal (A=B)if they have the
same number of rows and columns and if corresponding
entries are equal.
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Example
Given
Discuss the possibility that A=B, A=C and B=C
713
402A
13
02
B
dc
baC
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Example
A=B(impossible because A and B are different
size)
A=C(impossible because A and C are different
size)
B=C (possible provided that corresponding
entries are equal)
713402A
13
02B
dc
baC
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Matrix Addition
If A and B are 2 matrices of the same size,
A + B is defined to be the matrix of the same size formed
by adding corresponding entries.
If A=[aij] and B=[bij],[aij]+[bij]=[aij+bij]
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Matrix Addition
A =a1,1 a1,2 a1,3a2,1 a2,2 a2,3
B =b1,1 b1,2 b1,3b2,1 b2,2 b2,3
A+B=a1,1 a1,2 a1,3a2,1 a2,2 a2,3
b1,1+b2,1+
b1,2+
b2,2+
b1,3+b2,3+
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Matrix Subtraction
A-B is defined by subtracting corresponding entries.
If A=[aij] and B=[bij],
[aij]-[bij]=[aij- bij]
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Matrix Subtraction
A =a1,1 a1,2 a1,3a2,1 a2,2 a2,3
B =b1,1 b1,2 b1,3b2,1 b2,2 b2,3
A-B=a1,1 a1,2 a1,3a2,1 a2,2 a2,3
b1,1-b2,1-
b1,2-
b2,2-
b1,3-b2,3-
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Example
415
023A
357
241B
1412
262
34)5(175
2042)1(3BA
762
224
34)5(175
2042)1(3BA
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Theorem
If A, B and C denote arbitrary mxn
matrices, then,
A+B=B+A (commutative law)A+(B+C)=(A+B)+C (associative law)
0+A=A (0 is the mxn zero matrix)
A+(-A)= 0
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Scalar Multiplication
If Ais a matrix and cis a number, the scalar product
cAis the matrix formed from Aby multiplying each entry
of Aby the number c.
If A=[aij] ; cA=c[aij] a1,1 a1,2 a1,3A = a2,1 a2,2 a2,3
cA =ca1,1ca1,2ca1,3ca2,1ca2,2ca2,3
cA =a1,1 a1,2 a1,3a2,1 a2,2 a2,3
c
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Example
415
023A
357
241B
8210046
41502322A
171711
689
8210
046
91521
612323 AB
91521
6123
357
24133B
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Theorem
Let Aand Bdenote matrices and let c
and ddenote numbers,
c(A+B)=cA+cB(c+d)A=cA+dA
c(dA)=(cd)A
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Theorem
If cA=0, then either c=0 or
A=0
c(0)=0
0A=0
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Matrix Multiplication
If Ais an mxnmatrix and Bis an nxpmatrix
The number of columns of the left matrix (n)
must be same as the number of rows of the right
matrix (n)
Their matrix product (AB) is the mxpmatrix
whose (i,j) entry is the dot product of row iof A
and columnjof B.
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Matrix Multiplication
Matrix of size 2x3Matrix of size 3x2
zyx
yx
z
y
x
45
23
415
023
Matrix of size 2x3 Matrix of size 3x1 Matrix of size 2x1
cba
ba
zyx
yx
c
b
a
z
y
x
45
23
45
23
415
023
Matrix of size 2x2
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Transposition
Given an mxn matrix A, the transposeof
Ais the nxmmatrix, denoted by AT, whose
columns are formed from the
corresponding rows of A.
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Example
415
023A
40
12
53TA
zy
xwB
zx
ywBT
3
2
1
TC
321C
Row
Column
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Theorem
Let Aand Bdenote matrices whose
sizes are appropriate for the following
sums and products.(AT)T=A
(A+B) T=AT+BT
For any scalar r, (rA)T=rAT
(AB)T=BTAT
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Exercise
315
012A
8117
204B
31
40
12
C
(a) A + B (b) 3A-B (c) AC (d) A-2B
(e) AT+C (f) CB (g) AT+BT (h) CT-B
(i) AAT (j) ATA (k) ATB (l) CT-A
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Determinants
The determinant of a 2x2 matrix
For 1x1 matrix A=[a], |A|=a
bcaddcbaAA
det
dc
baA
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Example
Compute the determinant of
45
12A
3585*14*245
12detdet
A
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Submatrix
For any square matrix A, let Aijdenote the submatrix
formed by deleting the i-th row and j-th column of A.
Example:
987654
321
A
97
64
12A
C f
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Cofactors
Given A=[aij]
The(i,j)-cofactor of Ais Ci,jgiven by
ij
ji
ij AC det)1(
The + or sign in the
(i,j)-cofactor dependson the position of aijinthe matrix.
(-1)1+1
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Determinants of n 2
nn CaCaCaA 1112121111 ...det
D i f 2
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Determinants of n 2
For n2, the determinant of an nn
matrixA=[aij] is the sum of n terms of the
form a1jdet A1j, with plus and minussigns alternating, where the entries a11,
a12, ,a1nare from the first row of A.
AaAaAaA nn det)1(..detdet 1
112121111
D t i t
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Determinants
The determinant of an nn matrix A can
be computed by a cofactor expansion
across any row or down any column.
D t i t
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Determinants
The expansion across the i-th row using the cofactor
The cofactor expansion down the j-th column is
ininiiii CaCaCaA ...det 2211
njnjjjjj CaCaCaA ...det 2211
D t i t
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Determinants
The determinant of a 3x3 matrix using first row
expansion:
333231
232221
131211
aaa
aaa
aaa
A
3231
2221
13
3331
2321
12
3332
2322
11 detdetdetdet
aa
aaa
aa
aaa
aa
aaaA
E l
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Example
Use a cofactor expansion across the 3rd row to compute
det A.
l
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example
E i
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Exercise
M t i I
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Matrix Inverses
If Ais a square matrix, a matrix Cis
called an inverse of Aif,
AC= I and CA=IC=A-1
M t i I
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Matrix Inverses
E l
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Example
Matri In erses (3 3)
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Matrix Inverses (3 x 3)
Using cofactor matrix to compute the inverse:
TAA
AA
A )trixcofactorma(||
1)adjoint(||
11
E l
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Example
Det (A) =|A| =
A=Find the inverse of the matrix
Solution:
1. Find the determinant of A (in this case, using Row 1)
Example(cont )
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Example(cont.)
A12
A13=
A11= (-1+2) =1
A21=
A22=
A23=
A31=
A32=
A33=
2. Find the determinant of
submatrix (minor of entry ai j):
Example(cont )
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Example(cont.)
3. Find the adjoint of matrix A:
TAA )trixcofactorma(Adjoint
3. Find the cofactors of matrix A:
141
183
121
cofactorsofmatrix
141
183
121
111
482
131
Example(cont )
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100
A-1 =
4. Calculate the inverse:
Example(cont.)
)adjoint(||
11A
A
A
Matrix Inverses using Linear System
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Matrix Inverses using Linear System
Suppose Ais a square matrix and there exists a
sequence of elementary row operations that
carry AI. Then Ais invertible and this same
sequence carries IA-1.
[ A I] [ I A-1]
where the row operations on A and Iare carried out
simultaneously.
Example
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Example
Solution:
Solution (cont )
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Solution (cont.)
Solution (cont )
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Solution (cont.)
Solution (cont )
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Solution (cont.)
Solution (cont )
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Solution (cont.)
Solution (cont )
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Solution (cont.)
Exercise
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Exercise
Find the inverse of the given matrices.
042
361
123
A
i) Using the Adjoint
i) Using the linear systems.
Matrix Factorization
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Matrix Factorization
The solution of a system of linear equations AX = B
can be computed much more quickly if the matrix A
can be factored in the form
A = LU
where
L is a lower triangular matrix
Uis an upper triangular matrix
Matrix Factorization
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Matrix Factorization
a a a
a a a
a a a
x
x
x
b
b
b
11 12 13
21 22 23
31 32 33
1
2
3
1
2
3
a a a
a a a
a a a
l
l l
l l l
u u u
u u
u
11 12 13
21 22 23
31 32 33
11
21 22
31 32 33
11 12 13
22 23
33
0 0
0 0
0 0
(Ax=B)
(A=LU)
Matrix Factorization
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Matrix Factorization
l u l u l u
l u l u l u l u l u
l u l u l u l u l u l u
a a a
a a a
a a a
11 11 11 12 11 13
21 11 21 12 22 22 21 13 22 23
31 11 31 12 32 22 31 13 32 23 33 33
11 12 13
21 22 23
31 32 33
l
l l
l l l
u u u
u u
u
a a a
a a a
a a a
11
21 22
31 32 33
11 12 13
22 23
33
11 12 13
21 22 23
31 32 33
0 0
0 0
0 0
Matrix Factorization
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Matrix Factorization
AX=Bcan be solved in 2 stages:
First solve LY=Bfor Yby forward substitution.
Then solve UX=YforXby back substitution.
Matrix Factorization
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Matrix Factorization
LY=B
Forward substitution
33
2321313
3
22
1212
2
11
1
1
,
,
l
ylylby
l
ylby
l
by
l
l l
l l l
y
y
y
b
b
b
11
21 22
31 32 33
1
2
3
1
2
3
0 0
0
Matrix Factorization
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Matrix Factorization
UX=Y
Back substitution
u u u
u u
u
x
x
x
y
y
y
11 12 13
22 23
33
1
2
3
1
2
3
0
0 0
11
31321211
22
32322
33
33
,
,
u
xuxuyx
u
xuyx
u
yx
Doolittle Factorization
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Doolittle Factorization
The diagonal elements of matrix Lare 1s.
nn
n
n
nnnnnn
n
n
u
uuuuu
ll
l
aaa
aaaaaa
ULA
..00
:..::
..0
..
1..
:..::
0..10..01
..
:..::
..
..222
11211
21
21
21
22221
11211
Doolittle Factorization
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Doolittle Factorization
nnnn
n
n
nn
n
n
nn aaa
aaa
aaa
u
uu
uuu
ll
l
..
:..::
..
..
..00
:..::
..0
..
1..
:..::
0..1
0..01
21
22221
11211
222
11211
21
21
Example
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Example
Find the LUfactorization of the matrix
using Doolittle form and use it to solve the
linear system.
14
11
8
241
113
121
3
2
1
x
x
x
Solution (cont.)
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Solution (cont.)
241
113
121
00
0
1
01
001
33
2322
131211
3231
21
u
uu
uuu
ll
l
111u 212 u 113 u
Solution (cont.)
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Solution (cont.)
241
113
121
00
0
121
1
01
001
33
2322
3231
21
u
uu
ll
l
313
21 l 11131 l
Solution (cont.)
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( )
241
113
121
00
0
121
11
013
001
33
2322
32 u
uu
l
5)2)(3(122 u 2)1)(3(123 u
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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( )
241
113
121
00
250
121
11
013
001
3332 ul
5
2
5
)2)(1(432
l
Solution (cont.)
8/12/2019 Linear Equation and Matrices
122/138
( )
241
113
121
00
250
121
11
013
001
3352 u
51)2(
52)1)(1(233
u
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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( )
241113
121
00250
121
11013
001
51
52
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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( )
14
11
8
11
013
001
3
2
1
52 y
y
y81y
13)8)(3(112 y
54
52
3 )13)(()8)(1(14 y
TY 54
138
Solution (cont.)
8/12/2019 Linear Equation and Matrices
125/138
( )
54
3
2
1
51
13
8
00
250
121
x
x
x 451
54
3 x
15
)4)(2(132 x
21
)4)(1()1)(2(8
1
x TX 412
Exercise
8/12/2019 Linear Equation and Matrices
126/138
Find the LUfactorization of the matrixAusing Doolittle
form and use it to solve the linear systemAX=B.
(a) B=[7 2 10]T (b) B=[23 35 7]T
321
921
611
A
Crout Factorization
8/12/2019 Linear Equation and Matrices
127/138
The diagonal elements of matrix Uare 1s.
ll l
l l l
u uu
a a aa a a
a a a
11
21 22
31 32 33
12 13
23
11 12 13
21 22 23
31 32 33
0 00
10 1
0 0 1
Example
8/12/2019 Linear Equation and Matrices
128/138
Find the LUfactorization of the matrix using Crout form
and use it to solve the linear system.
26
20
12
611
352
327
3
2
1
x
x
x
Solution
8/12/2019 Linear Equation and Matrices
129/138
333231
2221
11
0
00
lll
ll
l
L
100
10
1
23
1312
u
uu
U
Solution (cont.)
8/12/2019 Linear Equation and Matrices
130/138
611
352327
100
101
000
23
1312
333231
2221
11
uuu
lll
lll
LU=A
Solution (cont.)
8/12/2019 Linear Equation and Matrices
131/138
711 l 221 l 131 l
611
352
327
100
10
1
0
00
23
1312
333231
2221
11
u
uu
lll
ll
l
Solution (cont.)
8/12/2019 Linear Equation and Matrices
132/138
7
212u 7
313
u
611
352
327
100
10
1
1
02
007
23
1312
3332
22 u
uu
ll
l
Solution (cont.)
8/12/2019 Linear Equation and Matrices
133/138
7
31
7
22522 l
31
15
27
33
7
31
23
23
u
u
611
352
327
100
10
7/37/21
1
02
007
23
3332
22 u
ll
l
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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7
9
7
2
132
l 31
192
31
)15(
7
)9(
7
3
633
l
611
352
327
100
31/1510
7/37/21
1
07/312
007
3332 ll
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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611
352
327
100311510
73721
31192791
07312
007
Solution (cont.)
8/12/2019 Linear Equation and Matrices
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26
20
12
31192791
07312
007
3
2
1
y
y
y
192
902
,31
116
,7
12
3
2
1
y
y
y
Solution (cont.)
8/12/2019 Linear Equation and Matrices
137/138
192902
31116712
100
311510737/21
3
2
1
x
xx
192
138
192
902
7
3
192
282
7
2
7
12
192
282
192
902
31
15
31
116
192902
1
2
3
x
x
x
Exercise
8/12/2019 Linear Equation and Matrices
138/138
Find the LUfactorization of the matrixAusing Crout
form and use it to solve the linear systemAX=B.
(a) B=[-4 10 5]T (b) B=[20 49 32]T
231
351
642
A