ce190 - lecture 5
TRANSCRIPT
NUMERICAL METHODSFOR ENGINEERS
LECTURE 05
LU DECOMPOSITION
AND
MATRIX INVERSION
SJSU, by Ngoc Pham + Udeme Ndon 1Lecture Note 5
LU DECOMPOSITION
Definition:
A nonsingular matrix A has a triangular factorization if it can be expressed as the product of a lower triangular matrix L and an upper triangular matrix U.
• Solve A . X = B (system of linear equations)
• Decompose A = L . U
*
L : Lower Triangular Matrix U : Upper Triangular Matrix
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 2
00
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 3
Solving Systems of equation by LU Decomposition:
Assume: [A]{X}={B}
[L][U]=[A] [L][U]{X}={B}
Consider [U]{X}={d}[L]{d}={B}
1.Solve [L]{d}={B} using forward substitution to get {d}
2.Use back substitution to solve [U]{X}={d} to get {X}
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 4
BXA BXUL
333231
232221
131211
aaa
aaa
aaa
A
[ U ][ L ]
33
2322
131211
3231
21
00
0
1
01
001
u
uu
uuu
ll
l
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 5
''
'
''
''
3
2
1
3
2
1
33
2322
131211
00
0
b
b
b
x
x
x
a
aa
aaa
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
Gauss Elimination
BXA BXUL
1
01
001
3231
21
ll
lL
Coefficients used during the elimination step
[ U ]
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon
Lecture Note 5 6
''
''
33
2322
131211
3231
21
333231
232221
131211
00
0
1
01
001
a
aa
aaa
ll
l
aaa
aaa
aaa
A
[ L . U ]
?32
11
3131
11
2121
l
a
al
a
al
LU DECOMPOSITION
Example 1: Solve the following system of equations using LU Decomposition.
2x - 3y + 4z = 5
X + 2y – 3z = 2
4x + y + 2z =1
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 7
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 8
LU DECOMPOSITION
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 9
INVERSE MATRIX
Inverse of a Matrix:
–If A is a square nxn matrix, and if there exist an nxnmatrix A-1 such that.
AA-1 = I
A-1 is called the inverse of the matrix A
Not all square matrix has its inverse. – If there is an inverse for a matrix A, then A is said to be a
nonsingular matrix or to be non singular.
– If A has no inverse, it is said to be a singular matrix or to be singular.
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 10
INVERSE MATRIX
For a matrix equation of the form:
AX = B
The following rules apply if A-1 exists:
• A-(AX) = A-B
• (A-A)X = A-B
• IX = A-B
• X = A-B
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 11
INVERSE MATRIX
To find an inverse of a matrix A, if it exists, we:
1. Form the matrix [A|I] as an augmented matrix
2. Reduce matrix A to an identity matrix while I becomes a non-identity matrix
3. The new non-identity nxn matrix formed from I is inverse of A (A-)
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 12
INVERSE MATRIX
Example 2: Find the inverse of matrix A
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 13
340
431
011
A
INVERSE MATRIX
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 14
INVERSE MATRIX
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 15
INVERSE MATRIX
Example 3: Show that matrix A has no inverse
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 16
32
64A
INVERSE MATRIX
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 17
INVERSE MATRIX
Matrix Inverse Method for a 2x2 matrix:
For a 2x2 matrix A, the inverse A-1 is given by:
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 18
1121
1222
21122211
1
2221
1211
1
aa
aa
aaaaA
aa
aaA
INVERSE MATRIX
Example: Find the inverse of matrix A
SJSU, by Ngoc Pham + Udeme Ndon Lecture Note 5 19
32
11
32
11
1
1
32
11
)2)(1()1)(3(
1
12
13
1
1
A
A
A