David J. Krus Presents

David J.
Krus
presents
Matrix Algebra
for Social Sciences
Introduction to
Matrix Algebra
Dimensions of a Matrix
 Number of Rows: 2
 Number of Columns:3
 A 2 x 3 Matrix
Elements of a Matrix
Principal Diagonal
Elements
Off-Diagonal Elements
Nomenclature of
Matrices
 Rectangular
 Square
 Symmetric
 Skew
Symmetric
Transpose
Triangulation
Matrix Algebra
Operations
on Matrix Elements
Addition of Matrix
Elements
2x2 2x2 2x2
 All matrices must have the

the same dimensions.
 The plus sign is enclosed in

parentheses.
Addition of Matrix
Elements
Subtraction of Matrix
Elements
2x2 2x2 2x2
 All matrices must have

the the same dimensions.
 The subtraction sign is

enclosed in parentheses.
Subtraction of Matrix
Elements
Multiplication of Matrix
Elements
2x2 2x2 2x2
 All matrices must have the

the same dimensions.
 The multiplication sign is

Multiplication of Matrix
Elements
Division of Matrix
Elements
 All matrices must have the the same
dimensions or the divisor must be a
scalar number.
 The division sign is enclosed in

parentheses.
2x2 2x2 2x2

Division of Matrix
Elements
Powers of Matrix
Elements
 The square sign is
Powers of Matrix
Elements
 The square sign is

enclosed in parentheses
Matrix Algebra
Operations
on Matrices
Addition of Matrices
3x1 1x3 3x3

Major Addition of
Matrices
1+1=2
1+2=3
1+3=4
Major Addition of
Matrices
2+1=3
2+2=4
2+3=5
Major Addition of
Matrices
3+1=4
3+2=5
3+3=6
Minor Addition of
Matrices
(1+1) + (2+2) + (3+3) = 12

Subtraction of
Matrices
1x3 3x1 1x1

Minor Subtraction of
Matrices
(1-1) + (2-2) + (3-3) =0

Major Subtraction of
Matrices
1-1=0
1 - 2 = -1
1 - 3 = -2
Matrices
2-1=1
2-2=0
2 - 3 = -1
Matrices
3-1=2
3-2=1
3-3=0
Multiplication of
Matrices
3x2 2x3 3x3

Multiplication of
Matrices
(1*7) + (2*10) =27

(1*8) + (2*11) =30
(1*9) + (2*12) =33
Multiplication of
Matrices
(3*7) + (4*10) = 61
(3*8) + (4*11) = 68
(3*9) + (4*12) = 75
Multiplication of
Matrices
(5*7) + (6*10) = 95
(5*8) + (6*11) = 106
(5*9) + (6*12) = 117
Matrix Inversion
Matrix Inversion
Matrix Inversion
Powers of Matrices
Powers of Matrices
(1*1) + (2*3) = 7 (1*2) + (2*4) = 10

(3*1) + (4*3) = 15 (3*2) + (4*4) = 22
Elements Of Statistics
Algebraic Mean
In Summation Notation
Summation Notation
X
MX 
n
Algebraic Mean
In Matrix Algebra Notation

Matrix Algebra Notation
1' X
Mx 
n
Matrix Multiplication
1 
 2
 
1 1 1 1 1 3
 
 4
5 15
Mx   3
5 5
Mean
1 
 2
 
1 1 1 1 1 3
 
 4
5 15
Mx   3
5 5
True Variance
Summation Notation
nX  (X )
2 2
 
2
x 2
n
True Variance

1' ( X  X ' ) 1( 2)
x 
2
2

n
Matrix Subtraction: X – X’
( 2)
 1   1
   1
 2  
1 1 1 1 1  3  1 2 3 4 5  1
  
 4  1
   1
 5  
x 
2 
52
Resulting Pairwise Differences
( 2)
0  1  2  3  4 1
1 0  1  2  3 1
 
1 1 1 1 1 2 1 0  1  2 1
  
3 2 1 0  1 1
4 3 2 1 0   1
 x2  2
5
Triangulate the Matrix
( 2)
0 0 0 0 0 1
1  
0 0 0 0 1 

1 1 1 1 1 2 1 0 0 0 1
  
3 2 1 0 0 1

 4 3 2 1 0 1
x 
2
52
Square the Matrix Elements
0 0 0 0 0 1
1 0  
0 0 0 1 

1 1 1 1 1  4 1 0 0 0 1
  
9 4 1 0 0 1

16 9 4 1 0 1
x 
2
25
Variance
Sum the
50 squared
 Relational 2
2 elements
x
25
space
Covariance
Summation Notation
xy
cov xy 
n
Covariance

D D
C
n
Obtained Scores
2 1
1 2 
 
X  5 3
 
 4 4

3 5
Deviation Scores
x y
1  2
 2 1 
 
D  2 0
 
 1 1

 0 2
D D
C
n
  1  2
 2  1
  1  2 2 1 0  
  2  1 0 1 1  2 0
  
 1 1  10 5 
 0 2   5 10
C     2 1
 1 2
5 5  
Diagonal Elements:
Sums of Squares
  1  2
 2  1
  1  2 2 1 0  
  2  1 0 1 1  2 0 
   x’x 5 
 1 1 10
 0 2   5 y’y 
 10 2 1
C   
5 5 1 2
Off-Diagonal Elements:
Cross-Products
  1  2
 2  1
  1  2 2 1 0  
  2  1 0 1 1  2 0 
  
 1 1  10 5
xy
 0 2  yx 
5 10 2 1
C   
5 5 1 2 
Variance-Covariance Matrix
  1  2
 2  1 
  1  2 2 1 0  
  2  1 0 1 1  2 0 
  
 1 1  10 5 
 0 2   5 10 2 1
C   
5 5 1 2 
Correlation
Summation Notation
z x z y
rxy 
n
Correlation

Z Z
R
n
Obtained Scores
2 1
1 2 
 
X  5 3
 
 4 4

3 5
Standard Scores
Zx Zy
 .71  1.41
  1.41 .71 
 
Z   1.41 .00
 
 .71 .71
 .00 1.41
Matrix Multiplication: Z’Z
 .71  141
. 
 141
. .71
 .71  141
. 141
. .71 .00  
 141   141
. .00
 . .71 .00 .71 141
.  
 .71 .71
 .00 . 
141
R
5
Resulting Matrix
ZxZx
5.0 2.5
ZxZy
ZyZx
2.5 5.0
R  ZyZy
5
Correlation Matrix
100
. .50
R 
 .50 100
. 

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David J.

2x2 2x2 2x2

 All matrices must have the

 The plus sign is enclosed in

2x2 2x2 2x2

 All matrices must have

 The subtraction sign is

2x2 2x2 2x2

 All matrices must have the

 The multiplication sign is

 The division sign is enclosed in

2x2 2x2 2x2

 The square sign is

3x1 1x3 3x3

(1+1) + (2+2) + (3+3) = 12

1x3 3x1 1x1

(1-1) + (2-2) + (3-3) =0

3x2 2x3 3x3

(1*7) + (2*10) =27

(1*1) + (2*3) = 7 (1*2) + (2*4) = 10

In Matrix Algebra Notation

In Matrix Algebra Notation

In Matrix Algebra Notation

In Matrix Algebra Notation

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(17) + (210) =27

(11) + (23) = 7 (12) + (24) = 10