Math 2
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Transcript of Math 2
Bowen’s MathematicsChapter 1, 2
Systems of Linear Equations and Matrices
MATRIX: Definitions
2
Matrix: a rectangular array of numbers which we treat as a single (collective) object Demarcated within brackets, parentheses, or
double lines Denoted by bold capital letters Matrix has m rows and n columns
Definitions
3
Matrix: a rectangular array of numbers which we treat as a single (collective) object
Order of matrix = row X column Element of a matrix = element row column
Notation
4
Matrices are usually denoted using upper-case letters, while the corresponding lower-case letters, with two subscript indices, represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, e.g.,.
B
Definitions
5
=
333231
232221
131211
33
aaa
aaa
aaa
A X
MatrixShort notation
Order = 3x3
This is the 3,2 element, or element32
Types of Matrices
Square Matrix; Row Matrix; Column Matrix; Diagonal Matrix; Unit Matrix; Zero Matrix; Scalar Matrix; Sub Matrix; Symmetric Matrix
6
Definition
7
A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. A square matrix A is called invertible or non-singular if there exists a matrix B such that
AB = In.
Definitions
8
Vector: a matrix containing a single row (row vector) or single column (column vector)
Square matrix: same # rows and columns
Definitions
9
Diagonal matrix: one or more non-zero values on main diagonal (top-left to bottom-right); all zeros on off-diagonal
800
050
001
Definitions
10
Scalar matrix: diagonal matrix with same non-zero values on entire main diagonal
500
050
005
Definitions
11
Unity or identity matrix is special case of scalar matrix where diagonal values = 1
100
010
001
Definitions
Symmetric Matrix: matrix remains unchanged when rows and columns are interchanged
511
162
125
12
Definitions
13
Equality: two matrices of same order and with all equal elements
Assume and
Then A = B implies that x=2 and y=3
=23
21A =
2
1
y
xB
Matrix Operations
14
In matrix algebra, an ordinary number is called a “scalar”.
In scalar multiplication, multiply all entries in matrix by the scalar.
The scalar often known as “scaling factor”
4872
12-36
46
1-312 =×
Addition and Subtraction
15
?127
264
354
234
?27
64
354
234
=±
=±
Some Laws of Matrix
16
Commutative LawA + B = B + A
Associative LawA + (B + C) = (A + B) + C
Identity LawA + O = AA – O = AO – A = -A
O is zero matrix. In a zero matrix all entries are 0s.
Matrix Multiplication
17
Scalar can be used to multiply a matrix of any dimension
Multiplication of two matrices is contingent upon the satisfaction of different dimensional requirementGiven two matrices A and B and we want to
find AB. The conformability condition for multiplication is that the column of A (lead matrix) must be equal to the row dimension of B (lag matrix)
The Rule of Multiplication
18
231213112212121121121111
232221
131211
1211
][
babababababaABC
bbb
bbbB
aaA
pm
CBA mqpqmn
+++==
=
=
=
=×
Example
19
?39
25
04
82
31
=×== BABA
Laws
20
A commutative law for matrix multiplication does not hold.A • B ≠ B • A
An associative law for matrix multiplication does holdA • (B • C) = (A • B) • C = A • B • C
The product of any matrix and the zero matrix is zero matrixA • O = O • A = O
Identity Matrix
21
A • I = A and I • A = AThe role of identity (I) matrix is analogous to the role of
the number 1.
=++
++=×
50
12
)51()10()01()20(
)50()11()00()21(
50
12
10
01
xxxx
xxxx
Problem Set 2-4
22
63
52
41
110
010
001
.17
1-0
32
11
65
412-1-
2031
0112
.15
Problem 21
23
Interest at the rates 0.06, 0.07, and 0.08 is earned on respective investments of $3000, $2000 and $4000. Compute the total interest by matrix multiplication.
Problem 22
24
Two canned meat spreads, Regular and Superior, are made by grinding beef, pork and lamb together. The numbers of pounds of each meat in a 15-pound batch of each brand are in the following table.
Suppose we wish to make 10 batches of Superior and 20 of Regular. Multiply the meat matrix in the table and the batch vector ( 10 20) and interpret the result.
Suppose that the per pound prices of beef, pork, and lamb are $2.50, $2.00, and $3.00, respectively. Multiply the price vector and the meat matrix and interpret the results.
Brand Pounds of
Beef Pork Lamb
Superior 8 2 5
Regular 4 8 3
Basic Operations
25
Transposition of vector or matrix from A to A': a11 → a'11; a21 → a'12; a12 → a'21; etc. From A to A'
db
ca
dc
ba→
Transpose
Transpose: Swap rows with columns
=
ihg
fed
cba
M
26
=
ifc
heb
gda
M T
=
z
y
x
V [ ]zyxV T =
Basic Operations
27
Transposition of vector or matrix from A to A': a11 → a'11; a21 → a'12; a12 → a'21; etc. From A to A'
db
ca
dc
ba→
Transpose
Transpose: Swap rows with columns
=
ihg
fed
cba
M
28
=
ifc
heb
gda
M T
=
z
y
x
V [ ]zyxV T =
Transpose of a Matrix
29
A matrix obtained by interchanging rows and columns.
Inverse of a Matrix
30
IAA
aa
=
=
1-
1-
1-
-1
I.matrix identity theisproduct
their ifother each of inverses are matrices square Two
identity. theis a and a ofproduct the
because a of inverse tivemultiplica theis a
1)(
How to find Inverse of a Matrix?
31
Gauss-Jordan InversionThe Gauss-Jordan method finds A’. It
transforms the augmented matrix (A|I) into the augmented matrix (I|A’).
10
01
ba
yx
Example: Find Inverse
32
5-42
1-1-1
1-21
3
1
1-02
01-1
1-23
72-
3-1
12
37
1-
1-
==
==
AA
AA
Conditions of Inverse
33
It is a square matrixIt has independent rows and columnsDeterminant of the matrix is not zero
What is determinants?
34
A “useful number” associated with an n n matrix( ) [ ] ( )
1111 det , If 1 aAaA ==
( ) ( )21122211
2221
1211-det , If 2 aaaaA
aa
aaA ==
( )( )
( )jjjjjj
ininiiii
CaCaCaA
or
CaCaCaA
n
222211
2211
det
det
: 2For 3
+++=
+++=
>
35
( )( )
jjjjjj
ininiiii
CaCaCaA
CaCaCaA
222211
2211
det
det
+++=
+++=
Mij is the determinant obtained by deleting the ith row and jth column of A.
ijji
ij MC )1(
36
( ) ji+1-
++
++
++
++
--
--
--
--
++
+
++
-
--
-
The cofactor is the minor with a sign change considered
( )ij
jiij MC += 1-
Adjoint of a Square Matrix
37
Let A=[aij]nxn be a square matrix of order n, then adjoint of A is defined to be transpose of matrix [Aij]nxn, where Aij is co-factor of aij in |A|.
Example
38
41-3
132-
5-14
703
230
1-14
=
=
B
A
Inverse of a Matrix
Inverse of a square matrix is:
39
An n × n matrix is invertible if and only if
AdjAA
A11
0)det( A
Solution of Equation
40
1. Transform the equations in matrices Ax = d
2. A = coefficient matrix3. x = variable matrix4. d = RHS values5. Calculate the determinant of A …….|A|6. Calculate the cofactor matrix7. Calculate the Adj. A (Transpose of cofactor
matrix)8. Find the inverse of a matrix A-1= Adj A/|A|9. Solve x = A-1d
Example
41
• A manufacturer produces three products: P, Q and R which he sells in two markets. Annual sales volume are indicated as follows:
Market Product .
. P Q R .
I 10,000 2,000 18,000
II 6,000 20,000 8,000
. .
(i) If unit sale prices of P, Q, and R are Tk. 2.50, 1.25 and 1.50 respectively, find the total revenue in each market with the help of matrix algebra.
(ii)If the unit costs of the above 3 commodities are Tk. 1.80, 1.20 and 0.80 respectively, find the gross profit.
Basic Applications
42
Consider 2 equations, 2 unknowns:2x + 3y = 53x + 2y = 5
Now define the following column vectors:
Then the equations can be written asxa + yb = c
===5
5,
2
3,
3
2cba
Basic Applications
43
Why this works:Recall matrix (vector) addition rule
And matrix (vector) equality rule2x + 3y = 53x + 2y = 5
=+
+=+
5
5
)23(
)32(
2
3
3
2
yx
yx
y
y
x
x
Equation System to Matrix
44
4-3
92-4
6
=+
=+
=++
zyx
zyx
zyx
Inverse of a Matrix
Inverse of a square matrix is:
AadjA
A11 =
45
An n × n matrix is invertible if and only if
Solution of Equation
46
1dAx
dAx
=
=1. Transform the equations in
matrices Ax = d2. A = coefficient matrix3. x = variable matrix4. d = RHS values5. Calculate the determinant of
A …….|A|6. Calculate the cofactor matrix7. Calculate the Adj. A
(Transpose of cofactor matrix)
8. Find the inverse of a matrix A-1= Adj A/|A|
9. Solve x = A-1d
Example
47
114x-3
295x
63x-2
21
21
21
=
=+
=
x
x
x
Markov Process
48
• Markov Property: The state of the system at time t+1 depends only on the state of the system at time t
[ ] [ ] x | X x X x x X | X x X tttttttt ===== ++++ 111111 PrPr
• Stationary Assumption: Transition probabilities are independent of time (t)
49
Weather:
• raining today 40% rain tomorrow
60% no rain tomorrow
• not raining today 20% rain tomorrow
80% no rain tomorrow
Markov ProcessSimple Example
rain no rain
0.60.4 0.8
0.2
Stochastic FSM:
=8.02.0
6.04.0P
50
Weather:
• raining today 40% rain tomorrow
60% no rain tomorrow
• not raining today 20% rain tomorrow
80% no rain tomorrow
Markov ProcessSimple Example
• Stochastic matrix:Rows sum up to 1
• Double stochastic matrix:Rows and columns sum up to 1
The transition matrix:
51
– Gambler starts with $10
- At each play we have one of the following:
• Gambler wins $1 with probability p
• Gambler looses $1 with probability 1-p
– Game ends when gambler goes broke, or gains a fortune of $100
(Both 0 and 100 are absorbing states)
0 1 2 99 100
p p p p
1-p 1-p 1-p 1-pStart (10$)
Markov ProcessGambler’s Example
52
• Markov process - described by a stochastic FSM
• Markov chain - a random walk on this graph
(distribution over paths)
• We can ask more complex questions, like
Markov Process
[ ] ?Pr 2 ===+ b a | X X tt
0 1 2 99 100
p p p p
1-p 1-p 1-p 1-pStart (10$)
53
• Given that a person’s last cola purchase was Coke, there is a 90% chance that his next cola purchase will also be Coke.
• If a person’s last cola purchase was Pepsi, there is an 80% chance that his next cola purchase will also be Pepsi.
coke pepsi
0.10.9 0.8
0.2
Markov ProcessCoke vs. Pepsi Example
=8.02.0
1.09.0P
transition matrix:
54
Given that a person is currently a Pepsi purchaser, what is the probability that he will purchase Coke two purchases from now?
Pr[ Pepsi?Coke ] =
Pr[ PepsiCokeCoke ] + Pr[ Pepsi Pepsi Coke ] =
0.2 * 0.9 + 0.8 * 0.2 = 0.34
==66.034.0
17.083.0
8.02.0
1.09.0
8.02.0
1.09.02P
Markov ProcessCoke vs. Pepsi Example (cont)
Pepsi ? ? Coke
=8.02.0
1.09.0P
55
Given that a person is currently a Coke purchaser, what is the probability that he will purchase Pepsi three purchases from now?
Markov ProcessCoke vs. Pepsi Example (cont)
==562.0438.0
219.0781.0
66.034.0
17.083.0
8.02.0
1.09.03P
Markov ProcessCoke vs. Pepsi Example (cont)
56
=8.02.0
1.09.0P
•Assume each person makes one cola purchase per week
•Suppose 60% of all people now drink Coke, and 40% drink Pepsi
•What fraction of people will be drinking Coke three weeks from now?
=562.0438.0
219.0781.03P
Pr[X3=Coke] = 0.6 * 0.781 + 0.4 * 0.438 = 0.6438
Qi - the distribution in week i
Q0=(0.6,0.4) - initial distribution
Q3= Q0 * P3 =(0.6438,0.3562)
Markov ProcessCoke vs. Pepsi Example (cont)
57
Simulation:
week - i
Pr[X
i = C
oke]
2/3
[ ] [ ]31
32
31
32
8.02.0
1.09.0=
stationary distribution
coke pepsi
0.10.9 0.8
0.2
Steady State
58
A steady state vector is the state vector that remains unchanged by the transition matrix[v1 v2] X [T] = [v1 v2]
Example
59
1
4.06.0
2.08.0
21
21
=+
×
vv
vv
What is steady state matrix?
Example
60
Absorbing MarkovStationary Markov
4.6.
01