chapter one matrices and system equations objective:to provide solvability conditions of a linear...
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CHAPTER ONECHAPTER ONEMatrices and System EquationsMatrices and System Equations
Objective:To provide solvability conditions of a linear equation Ax=b and introduce the Gaussian elimination method, a systematical approach in solving Ax=b, to solve it.
OutlineOutline
Motivative Example.Elementary row operations and Elementary M
atrices.Some Basic Properties of Matrices.Gaussian Elimination for solving Ax=b.Solvability conditions for Ax=b.
Motivative Example (curve fittingMotivative Example (curve fitting)) Given three points( )( )( ),find a
polynomial of degree 2 passing through the three given points.
Solution: Let the polynomial be
Where a,b and c are to be determined
Ax=b
11, yx 22 , yx 33, yx
cbxaxxy 2)(
c
b
a
xx
xx
xx
y
y
y
1
1
1
323
222
121
3
2
1
QuestionQuestion: Why transform to matrix : Why transform to matrix form?form?
To provide a systematic approach and to use computer resource.
QuestionQuestion: How to solve Ax=b systematically?: How to solve Ax=b systematically?
One way is to put Ax=b in triangular form,which
can be easily solved by back-substitution.
Definition: A system is said to be in triangular form if in the k-th equation the coefficients of thee first (k-1) variables are all zero and the coefficient of xk is nonzero ( k = 1,…,n)
QuestionQuestion: How to put Ax=b in triangular form : How to put Ax=b in triangular form while leaving the solution set invariantwhile leaving the solution set invariant??
Solution: By elementary row operations as described
below.
Definition: Two systems of equations involing the same variables are said to be equivalent if they have the same solution set.
Before introducing elementary operation, we Before introducing elementary operation, we recall some definitions and notations.recall some definitions and notations.
(§ 1.3) Equality of two matrices. Multiplication of a matrix by a scalar. Matrix addition. Matrix multiplication. Identity matrix. Multiplicative inverse. Nonsingular and singular matrix. Transpose of a matrix.
Def. If and , then the
Matrix Multiplication ,
where .
Def. An (n × n) matrix A is said to be nonsingular
or invertible if there exists a matrix B such that
AB=BA=I. The matrix B is said to be a
multiplicative inverse of A. And B is denoted
by A-1.Warning: In general, AB≠BA. Matrix multiplication is not commutative.
DefinitionsDefinitions
( ) m nijA a F ( ) n r
ijB b F
( ) m rijAB C c F
1
( ,:)n
ij j ik kjk
c a i b a b
Def. The transpose of an (m × n) matrix A is the (n ×
m) matrix B defined by for j=1,…,n and
i=1,…,m. The transpose of A is denoted by AT.
Def. An (n × n) matrix A is said to be symmetric if AT=A .
Definitions (cont.)Definitions (cont.)
ji ijb a
Some Matrix PropertiesSome Matrix Properties
Let be scalars,A,B and C be matrices
with proper dimensions.
(Commutative Law)
(Associative Law)
(Associative Law)
(Distributive Law)
(Distributive Law)
&
BCACCBA
ACABCBA
BCACAB
CBACBA
ABBA
)(
)(
)()(
)()(
TTT
TT
TT
BABA
AA
AA
BABA
AAA
BABAAB
AA
)(
)(
)(
)(
)(
)()()(
)()(
111)(
)(
ABAB
ABAB TTT
Some Matrix Properties (cont.)Some Matrix Properties (cont.)
NotationsNotations
,
The matrix is called an
augmented matrix.
In general, or .
n
n
nm
mnm
n
F
x
x
X
F
aa
aa
A
1
1
111
m
m
F
b
b
b
1
mmnm
n
baa
baa
bA
1
1111
F CF
Moreover,we define
1
1
( ,:)
(:, )
i in
j
j
mj
a i a a
a
a a j
a
1 2,
1
(1,:)
,
( ,:)
(1,:)
( ,:)
n
n
i ii
a
A a a a
a m
a x
Ax x a
a n x
Def: Let and .Then
is said to be a linear combination of .
Note that .We have the next result.
Theorem1.3.1: Ax=b is consistent b can be written
as a linear combination of colum vectors
of A.
1 2a ,a ,..., a nn F
1 2c ,c ,..., cn F1
am
i ii
c
i iAx x a
1 2a ,a ,..., an
Application 1: Weight ReductionApplication 1: Weight Reduction
Table 1
Calories Burned Per HourWeight in lb
Exercise Activity 152 161 170 178
Walking 2 mph 213 225 237 249
Running 5.5 mph 651 688 726 764
Bicycling 5.5mph 304 321 338 356
Tennis 420 441 468 492
Application 1: Weight Reduction (cont.)Application 1: Weight Reduction (cont.)
Table 2
Hours Per Day For Each ActivityExercise schedule
walking Running Bicycling Tennis
Monday 1.0 0.0 1.0 0.0
Tuesday 0.0 0.0 0.0 2.0
Wednesday 0.4 0.5 0.0 0.0
Thursday 0.0 0.0 0.5 2.0
Friday 0.4 0.5 0.0 0.0
Application 1: Weight Reduction Application 1: Weight Reduction (end)(end)
Solution:
1.0 0.0 1.0 0.0 605249
0.0 0.0 0.0 2.0 984764
0.4 0.5 0.0 0.0 481356
0.0 0.0 0.5 2.0 1162492
0.4 0.5 0.0 0.0 481.6
Application 2: Production CostsApplication 2: Production Costs
Table 3
Production Costs Per Item (dollars)Product
Expenses A B C
Raw materials 0.1 0.3 0.15
Labor 0.3 0.4 0.25
Overhead and
miscellaneous
0.1 0.2 0.15
Application 2: Production Costs (cont.)Application 2: Production Costs (cont.)
Table 4
Amount Produced Per QuarterSeason
Product Summer Fall Winter Spring
A 4000 4500 4500 4000
B 2000 2600 2400 2200
C 5800 6200 6000 6000
Application 2: Weight Reduction (cont.)Application 2: Weight Reduction (cont.)
Solution:0.1 0.3 0.15
0.3 0.4 0.25
0.1 0.2 0.15
M
4000 4500 4500 4000
2000 2600 2400 2200
5800 6200 6000 6000
P
Application 2: Weight Reduction (cont.)Application 2: Weight Reduction (cont.)
Solution:
1870 2160 2070 1960
3450 3940 3810 3580
1670 1900 1830 1740
MP
Application 2: Production Costs Application 2: Production Costs (end)(end)
Table 5
Amount Produced Per QuarterSeason
Summer Fall Winter Spring Year
Raw materials 1,870 2,160 2,070 1,960 8,060
Labor 3,450 3,940 3,810 3,580 14,780
Overhead and miscellaneous
1,670 1,900 1,830 1,740 7,140
Total production cost
6,990 8,000 7,710 7,280 29,980
Solution:
Application 5: Networks and Graphs Application 5: Networks and Graphs (cont.)(cont.)
n nIf A is a F ,
1 , if { , } is an edge of the graph.then
0 , if there is no edge joioning and .
i j
iji j
adjacency matrix
V Va
V V
for Figure 1.3.2,
0 1 0 0 0
1 0 0 0 1
adjacency matrix 0 0 0 1 1
0 0 1 0 1
0 1 1 1 0
A
DEF.
Application 5: Networks and Graphs Application 5: Networks and Graphs (end)(end)
3
0 2 1 1 0
2 0 1 1 4
1 1 2 3 4
1 1 3 2 4
0 4 4 4 2
A
Theorem 1.3.3. If A is an n × n adjacency matrix of a graph and represents the ijth entry of Ak, then is equal to the number of walks of length from to Vi to Vj.
( ) kija
( ) kija
Application 6: Information Retrieval (P.59)Application 6: Information Retrieval (P.59)
Suppose that our database, consists of these book titles:
B1. Applied Linear AlgebraB2. Elementary Linear AlgebraB3. Elementary Linear Algebra with ApplicationsB4. Linear Algebra and Its ApplicationsB5. Linear Algebra with ApplicationsB6. Matrix Algebra with ApplicationsB7. Matrix Theory
The collection of key words is given by the following alphabetical list:
algebra, application, elementary, linear, matrix, theory
Application 6: Information Retrieval (cont.)Application 6: Information Retrieval (cont.)
Table 8Array Representation for
Database of Linear Algebra Books
Books
Key Words B1 B2 B3 B4 B5 B6 B7
algebra 1 1 1 1 1 1 0
application 1 0 1 1 1 1 0
elementary 0 1 1 0 0 0 0
linear 1 1 1 1 1 0 0
matrix 0 0 0 0 0 1 1
theory 0 0 0 0 0 0 1
Application 6: Information Retrieval Application 6: Information Retrieval (end)(end)
If the words we are searching for are applied, linear, and algebra,then the database matrix and search vector are given by
1 1 1 1 1 1 0 1
1 0 1 1 1 1 0 1
0 1 1 0 0 0 0 0
1 1 1 1 1 0 0 1
0 0 0 0 0 1 1 0
0 0 0 0 0 0 1 0
A x
1 1 0 1 0 0 31
1 0 1 1 0 0 21
1 1 1 1 0 0 30
1 1 0 1 0 0 31
1 1 0 1 0 0 30
1 1 0 0 1 0 20
0 0 0 0 1 1 0
y
If we set y= ATx, then
Let’s back to solve Ax=bLet’s back to solve Ax=b Eg2
432
13
32
321
321
321
xxx
xxx
xxx
20
10670
32
32
32
321
xx
xx
xxx
7
4
7
1
1067
32
3
32
321
x
xx
xxx
4132
1113
3321
2110
10670
3121
7
4
7
100
10670
3121
(§ 1.2)
Three types of Elementary row operations.
I. Interchange two row.
II. Multiply a row by .
III. Replace a row by its sum with a multiple of
another row.
0\
Lead variables and free variables(p.15)Eg:
, and are lead variables while and
are free variables.
510000
223100
102021
3x1x 5x
4x
2x
Def. A matrix is said to be in row echelon form if
(i) The first nonzero entry in each row is 1.
(ii) If row k does not consist entirely of zero,
the number of leading zero entries in row
k+1 is grater then the number of leading
zero entries in row k.
(iii) If there are rows whose entries are all zero, they
are below the rows having nonzero entries.
Def. The process of using row operations I, II, and III to
transform a linear system into one whose augmented
matrix is in row echelon form is called Gaussian
elimination.
Def. A linear system is said to be overdetermined
if there are more equations(m) than unknowns
(n). (m > n) Warning: Overdetermined systems are usually (but not always) in consistent.
Def. A system of m linear equations in n unknowns
is said to be underdetermined if there are
fewer equations. (m < n)
Overdetermined and UnderdeterminedOverdetermined and Underdetermined
Def. A matrix is said to be in reduced row echelon
form if:
(i) The matrix is in row echelon form.
(ii) The first nonzero entry in each row is the
only nonzero entry in its column.
Def. The process of using elementary row operations
to transform a matrix into reduced row echelon
form is called Gauss-Jordan reduction.
Reduced Row Echelon FormReduced Row Echelon Form
Application 2: Electrical Networks Application 2: Electrical Networks (end)(end)
Kirchhoff’s Laws: 1. At every node the sum of the incoming currents equals the sum of the outgoing currents. 2. Around every closed loop the algebraic sum of the voltage must equal the algebraic sum of the voltage drops.
1 1 1 01 1 1 0
2 41 1 1 0 0 1
3 34 2 0 8
0 0 1 10 2 5 9
0 0 0 0
Application 4: Economic Models For Application 4: Economic Models For Exchange of Goods Exchange of Goods (P.25)(P.25)
F
M
C
1/2
1/4
1/4
F M C
1/3
1/3
1/3
1/2
1/4
1/4
(§ 1.4)(§ 1.4) Elementary MatricesElementary Matrices
Type I ( ): Obtained by interchanging rows i and j
from identity matrix.
Type II ( ): Obtained from identity matrix by
multiplying row i with .
Type III ( ): Obtained from identity matrix by adding
to row j.
ijE
)(iE
)(ijE
irow
means performing type I row operation on A. means performing type II row operation on A. means performing type III row operation on A.
means performing type I column operation on A. means performing type II column operation on A. means performing type III column operation on A.
AEij
AEi )(AEij )(
ijAE
( )iAE ( )ijAE
Elementary Row / Column OperationElementary Row / Column Operation
Theorem1.4.2:
If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.
With
The solution set of a linear equations is invariant under th
ree types row operation. and have the solution set.
)()(
)/1()(1
1
1
ijij
ii
ijij
EE
EE
EE
Ax b
EAx Eb
Def. A matrix B is row equivalent to A if there exists a
finite sequence of elementary matrices such that
Row Equivalent Row Equivalent (P.71)(P.71)
1 2, ,..., kE E E
1 1...k kB E E E A
Theorem1.4.3
(a) A is nonsingular.
(b) Ax=0 has only the trivial solution 0.
(c) A is row equivalent to I.
(a) (b) Let be a solution of Ax=0.
(b) (c) Let A ~ U, where U is in reduced row echelon form. Suppose U contains a zero row. by Th1.2.1, Ux=0 has a nontrivial solution thus A~I.
(c) (a)
A~I A= E1 …… Ek for some E1 … Ek
∵ each Ei is nonsingular. ∴ A is nonsingular. (by Th.1.2.1)
row
0x1 1 1
0 0 0( ) ( ) 0 0x A A x A Ax A
Proof of Theorem 1.4.3
Corollary1.4.4Corollary1.4.4 Ax=b has a unique solution A is nonsingular.Ax=b has a unique solution A is nonsingular.
Pf: " “ The unique solution is .
" " Suppose is the unique solution and A is
singular.
is also a solution of Ax=b.
A is nonsingular.
bAx 1
x̂
^^
^
)(
00
xZx
bZxA
AZZ
3.4.1Th
BUT in general, and AB=AC B=C.
Eg.
Moreover,AC=AB while .
01
01A
10
10B
11
10C
01
01
10
10BAAB
CB
AB6=BA
If A is nonsingular and row equivalent to I, so
there exists elementary matrices such that
then,
EEkk……EE11(A | I)= ((A | I)= (EEkk……EE11‧‧A | A | EEkk……EE11‧‧I) I) ( by )
= (I | = (I | EEkk……EE11‧‧I) I) ( by )
= (I | A= (I | A-1-1))
1 1
-11 1
... A = I ---------- 1
... = A ---------- 2
k k
k k
E E E
E E E I
1
2
Method For Computing
Q: Compute A-1 if .
Sol:
Example 4. (P.73)
1 4 3
1 2 0
2 2 3
A
1 4 3 1 0 0
1 2 0 0 1 0
2 2 3 0 0 1
A
1 1 11 0 0
2 2 21 1 1
0 1 04 4 41 1 1
0 0 16 2 6
A
1 1 1
2 2 21 1 1
4 4 41 1 1
6 2 6
A
Q: Compute A-1 if .
Sol:
Example 4. (cont.)
1 4 3
1 2 0
2 2 3
A
1 4 3 1 0 0
1 2 0 0 1 0
2 2 3 0 0 1
A
1 1 11 0 0
2 2 21 1 1
0 1 04 4 41 1 1
0 0 16 2 6
A
1 1 1
2 2 21 1 1
4 4 41 1 1
6 2 6
A
Diagonal and Triangular Matrices
Def. An n × n matrix A is said to be upper triangular if aij=0 for i > j and lower triangular if aij=0 for i > j.
Def. An n × n matrix B is diagonal if aij=0 whenever i ≠ j.
Triangular Factorization
If an n × n matrix C can be reduced to upper triangular form
using only row operation III, then C has an LU factorization.
The matrix L is unit lower triangular, and if i > j, then lij is the multiple of t he jth row subtracted from the ith row during the reduction process.
Example 6. (P.74)
2 4 2
1 5 2
4 1 9
A
1 0 0
11 0
22 3 1
2 4 2
0 3 1
0 0 8
L
U
row operation III
Mark:
2 4 2
1 5 2
4 1 9
LU A
Let A be an m × n matrix and B is an n × r matrix.
It is often useful to partition A and B and express the
product in terms of the submatrices of A and B.
In general, partition B into columns
then
partition A into rows , then
Block Multiplication
1( ,..., )rb b
1 2( , ,..., )rAB Ab Ab Ab
(1,:)
(2,:)
( ,:)
a
aA
a m
(1,:)
(2,:)
( ,:)
a B
a BAB
a m B
Case 1.
Case 2.
Case 3.
Block Multiplication (cont.)
1 2 1 2A B B AB AB
1 1
2 2
A ABB
A A B
11 2 1 1 2 2
2
BA A AB A B
B
Case 4.
Let
then
Block Multiplication (cont.)
11 1 11 1
1 1
and B t r
s t t r
s st t tr
A A B B
A F F
A A B B
11 1
11
, where r t
s rij ik kj
ks sr
C C
AB C F C A B
C C
Example 2. (P.85)
Let A be an n × n matrix of the form ,
where A11 is a k × k matrix (k < n ) . Show that A is nonsingular if and only if A11 and A22 are nonsingular.
11
22
A OA
O A
Solution:
Give two vectors ,
This product is referred to as a scalar product or an
inner product.
Scalar / Inner Product
and in nx y R
1
2 1 11 2 1 1 2 2( , ,..., ) T
n n n
n
y
yx y x x x x y x y x y R
y
Give two vectors ,
The product is referred to as the outer product
of .
Outer Product
and in nx y R
1 1 1 1 2 1
2 2 1 2 2 2 1 2
1 2
( , ,..., )
n
nT n nn
n n n nn
x x y x y x y
x x y x y x yxy y y y R
x y x y x yx
Txy
and x y