david j. krus presents matrix algebra for social sciences
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
David J. KrusDavid J. Krus
presentspresents
Matrix AlgebraMatrix Algebrafor Social Sciencesfor Social Sciences
Introduction toIntroduction to
Matrix AlgebraMatrix Algebra
Dimensions of a MatrixDimensions of a Matrix
Number of Rows: 2Number of Rows: 2 Number of Columns:3Number of Columns:3 A 2 x 3 MatrixA 2 x 3 Matrix
Elements of a MatrixElements of a Matrix
Principal Diagonal Principal Diagonal Elements Elements
Off-Diagonal Elements Off-Diagonal Elements
Nomenclature of Nomenclature of MatricesMatrices
RectangularRectangular
SquareSquare
SymmetricSymmetric
Skew Skew SymmetricSymmetric
TransposeTranspose
TriangulationTriangulation
Matrix Algebra Matrix Algebra OperationsOperations
on Matrix Elements
Addition of Matrix Addition of Matrix ElementsElements
All matrices must have the All matrices must have the the same dimensions.the same dimensions.
The plus sign is enclosed in The plus sign is enclosed in parentheses.parentheses.
2 x 2 2 x 2 2 x 2
Addition of Matrix Addition of Matrix ElementsElements
Subtraction of Matrix Subtraction of Matrix ElementsElements
All matrices must have All matrices must have the the same dimensions.the the same dimensions.
The subtraction sign is The subtraction sign is enclosed in parentheses.enclosed in parentheses.
2 x 2 2 x 2 2 x 2
Subtraction of Matrix Subtraction of Matrix ElementsElements
Multiplication of Matrix Multiplication of Matrix ElementsElements
All matrices must have the All matrices must have the the same dimensions.the same dimensions.
The multiplication sign is The multiplication sign is enclosed in parentheses.enclosed in parentheses.
2 x 2 2 x 2 2 x 2
Multiplication of Matrix Multiplication of Matrix ElementsElements
Division of Matrix Division of Matrix ElementsElements
All matrices must have the the same All matrices must have the the same dimensions or the divisor must be a dimensions or the divisor must be a scalar number.scalar number.
The division sign is enclosed in The division sign is enclosed in parentheses. parentheses.
2 x 2 2 x 2 2 x 2
Division of Matrix Division of Matrix ElementsElements
Powers of Matrix Powers of Matrix ElementsElements
The square sign is The square sign is enclosed in parenthesesenclosed in parentheses..
Powers of Matrix Powers of Matrix ElementsElements
The square sign is The square sign is enclosed in parenthesesenclosed in parentheses
Matrix Algebra Matrix Algebra OperationsOperations
on Matrices
Addition of MatricesAddition of Matrices
3 x 1 1 x 3 3 x 3
Major Addition of Major Addition of MatricesMatrices
1 + 1 = 21 + 2 = 31 + 3 = 4
Major Addition of Major Addition of MatricesMatrices
2 + 1 = 32 + 2 = 42 + 3 = 5
Major Addition of Major Addition of MatricesMatrices
3 + 1 = 43 + 2 = 53 + 3 = 6
Minor Addition of Minor Addition of MatricesMatrices
(1+1) + (2+2) + (3+3) = 12
Subtraction of Subtraction of MatricesMatrices
1 x 3 3 x 1 1 x 1
Minor Subtraction of Minor Subtraction of MatricesMatrices
(1-1) + (2-2) + (3-3) =0
Major Subtraction of Major Subtraction of MatricesMatrices
1 - 1 = 01 - 2 = -11 - 3 = -2
Major Subtraction of Major Subtraction of MatricesMatrices
2 - 1 = 12 - 2 = 02 - 3 = -1
Major Subtraction of Major Subtraction of MatricesMatrices
3 - 1 = 23 - 2 = 13 - 3 = 0
Multiplication of Multiplication of MatricesMatrices
3 x 2 2 x 3 3 x 3
Multiplication of Multiplication of MatricesMatrices
(1*7) + (2*10) =27
(1*8) + (2*11) =30
(1*9) + (2*12) =33
Multiplication of Multiplication of MatricesMatrices
(3*8) + (4*11) = 68(3*7) + (4*10) = 61
(3*9) + (4*12) = 75
Multiplication of Multiplication of MatricesMatrices
(5*7) + (6*10) = 95(5*8) + (6*11) = 106(5*9) + (6*12) = 117
Matrix InversionMatrix Inversion
Matrix InversionMatrix Inversion
Matrix InversionMatrix Inversion
Powers of MatricesPowers of Matrices
Powers of MatricesPowers of Matrices
(1*1) + (2*3) = 7 (1*2) + (2*4) = 10
(3*1) + (4*3) = 15 (3*2) + (4*4) = 22
Elements Of Elements Of StatisticsStatistics
Algebraic MeanAlgebraic Mean
In Summation NotationIn Summation Notation
Summation NotationSummation Notation
M XnX
AlgebraicAlgebraic Mean Mean
In Matrix Algebra NotationIn Matrix Algebra Notation
Matrix Algebra NotationMatrix Algebra Notation
nXM x
'1
Matrix MultiplicationMatrix Multiplication
35
155
54321
11111
xM
MeanMean
35
155
54321
11111
xM
True VarianceTrue Variance
In Summation NotationIn Summation Notation
Summation NotationSummation Notation
2
222 )(
nXXn
x
True VarianceTrue Variance
In Matrix Algebra NotationIn Matrix Algebra Notation
MatrixMatrix Algebra Notation Algebra Notation
2
)2(2 1)'('1
nXX
x
Matrix Subtraction: X – X’Matrix Subtraction: X – X’
2
)2(
2
511111
54321
54321
11111
x
Resulting Pairwise DifferencesResulting Pairwise Differences
2
)2(
2
511111
0123410123210123210143210
11111
x
Triangulate the MatrixTriangulate the Matrix
2
)2(
2
511111
0123400123000120000100000
11111
x
Square the Matrix ElementsSquare the Matrix Elements
2511111
01491600149000140000100000
11111
2
x
VarianceVariance
225502 x
Sum the squared elements
Relational space
CovarianceCovariance
In Summation NotationIn Summation Notation
Summation NotationSummation Notation
cov xyxyn
CovarianceCovariance
In Matrix Algebra NotationIn Matrix Algebra Notation
Matrix Algebra NotationMatrix Algebra Notation
C D Dn
Obtained ScoresObtained Scores
X
2 11 25 34 43 5
Deviation ScoresDeviation Scores
D
1 22 12 01 10 2
x y
Matrix MultiplicationMatrix Multiplication
C D Dn
Matrix MultiplicationMatrix Multiplication
C
1 2 2 1 0
2 1 0 1 1
1 2
2 1
2 0
1 1
0 2
5
10 5
5 10
5
2 1
1 2
Diagonal Elements:Diagonal Elements:Sums of SquaresSums of Squares
C
1 2 2 1 0
2 1 0 1 1
1 2
2 1
2 0
1 1
0 2
5
10 5
5 10
5
2 1
1 2
x’x
y’y
Off-Diagonal Elements:Off-Diagonal Elements:Cross-ProductsCross-Products
C
1 2 2 1 02 1 0 1 1
1 22 12 01 10 2
5
10 55 10
52 11 2
xy
yx
Variance-Covariance MatrixVariance-Covariance Matrix
C
1 2 2 1 02 1 0 1 1
1 22 12 01 10 2
5
10 55 10
52 11 2
CorrelationCorrelation
In Summation NotationIn Summation Notation
Summation NotationSummation Notation
rz znxyx y
CorrelationCorrelation
In Matrix Algebra NotationIn Matrix Algebra Notation
Matrix Algebra NotationMatrix Algebra Notation
R Z Zn
Obtained ScoresObtained Scores
X
2 11 25 34 43 5
Standard ScoresStandard Scores
Z
. .
. .
. .
. .. .
71 141141 71141 00
71 7100 141
Zx Zy
Matrix Multiplication: Z’ZMatrix Multiplication: Z’Z
R
. . . . .
. . . . .
. .
. .
. .
. .. .
71 141 141 71 00141 71 00 71 141
71 141141 71141 00
71 7100 141
5
Resulting Matrix Resulting Matrix
R
5 0 2 52 5 5 0
5
. .. .
ZxZx
ZyZy
ZxZy
ZyZx
Correlation MatrixCorrelation Matrix
R
100 5050 100. .. .