1565 matrix01-ppt
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Transcript of 1565 matrix01-ppt
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Matrix Algebra Basics
Pam PerlichUrban Planning 5/6020
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Algebra
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Matrix
A
a11 ,, a1n
a21 ,, a2n
am1 ,, amn
Aij
A matrix is any doubly subscripted array of elements arranged in rows and columns.
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Row Vector
[1 x n] matrix
jn aaaaA ,, 2 1
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Column Vector
i
m
a
a
aa
A 2
1
[m x 1] matrix
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Square Matrix
B 5 4 73 6 12 1 3
Same number of rows and columns
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The Identity
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Identity Matrix
I
1 0 0 00 1 0 00 0 1 00 0 0 1
Square matrix with ones on the diagonal and zeros elsewhere.
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Transpose Matrix
A'
a11 a21 ,, am1
a12 a22 ,, am 2
a1n a2n ,, amn
Rows become columns and columns become rows
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Matrix Addition and Subtraction
A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by:
Cij Aij Bij
Note: all three matrices are of the same dimension
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Addition
A a11 a12
a21 a22
B b11 b12
b21 b22
C a11 b11 a12 b12
a21 b21 a 22 b22
If
and
then
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Matrix Addition Example
A B 3 45 6
1 23 4
4 68 10
C
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Matrix Subtraction
C = A - BIs defined by
Cij Aij Bij
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Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]
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Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s
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Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d
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Computation: A x B = C
A a11 a12
a21 a22
B b11 b12 b13
b21 b22 b23
232213212222122121221121
2312131122121211 21121111
babababababababababababa
C
[2 x 2]
[2 x 3]
[2 x 3]
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Computation: A x B = C
A 2 31 11 0
and B
1 1 1 1 0 2
[3 x 2] [2 x 3]A and B can be multiplied
1 1 13 1 28 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*182*31*2 20*31*2 51*31*2
C
[3 x 3]
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Computation: A x B = C
1 1 13 1 28 2 5
12*01*1 10*01*1 11*01*1
32*11*1 10*11*1 21*11*182*31*2 20*31*2 51*31*2
C
A 2 31 11 0
and B
1 1 1 1 0 2
[3 x 2] [2 x 3]
[3 x 3]
Result is 3 x 3
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Inversion
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Matrix Inversion
B 1B BB 1 I
Like a reciprocal in scalar math
Like the number one in scalar math
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Linear System of Simultaneous Equations
1st Precinct : x1 x2 62nd Pr ecinct : 2x1 x2 9
First precinct: 6 arrests last week equally divided between felonies and misdemeanors.
Second precinct: 9 arrests - there were twice as many felonies as the first precinct.
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Solution
96
*1 21 1
2
1
xx
33
2
1
xx
1 21 1
Note: Inverse of is
1 21 1
96
*1 21 1
*1 21 1
* 1 21 1
2
1
xx Premultiply both sides by
inverse matrix
33
* 1 00 1
2
1
xx A square matrix multiplied by its
inverse results in the identity matrix.
A 2x2 identity matrix multiplied by the 2x1 matrix results in the original 2x1 matrix.
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aijxj bi or Ax bj1
n
x A 1Ax A 1b
n equations in n variables:
unknown values of x can be found using the inverse of matrix A such that
General Form
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Garin-Lowry Model
Ax y x
y Ix Axy (I A)x
(I A) 1 y x
The object is to find x given A and y . This is done by solving for x :
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Matrix Operations in Excel
Select the cells in which the answer will appear
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Matrix Multiplication in Excel
1) Enter “=mmult(“
2) Select the cells of the first matrix
3) Enter comma “,”
4) Select the cells of the second matrix
5) Enter “)”
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Matrix Multiplication in Excel
Enter these three key strokes at the same time:
control
shift
enter
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Matrix Inversion in Excel Follow the same procedure Select cells in which answer is to be
displayed Enter the formula: =minverse( Select the cells containing the matrix to be
inverted Close parenthesis – type “)” Press three keys: Control, shift, enter
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