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David J. KrusDavid J. Krus
presentspresents
Matrix AlgebraMatrix Algebrafor Social Sciencesfor Social Sciences
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Introduction toIntroduction to
Matrix AlgebraMatrix Algebra
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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
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Elements of a MatrixElements of a Matrix
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Principal Diagonal Principal Diagonal Elements Elements
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Off-Diagonal Elements Off-Diagonal Elements
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Nomenclature of Nomenclature of MatricesMatrices
RectangularRectangular
SquareSquare
SymmetricSymmetric
Skew Skew SymmetricSymmetric
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TransposeTranspose
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TriangulationTriangulation
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Matrix Algebra Matrix Algebra OperationsOperations
on Matrix Elements
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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
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Addition of Matrix Addition of Matrix ElementsElements
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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
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Subtraction of Matrix Subtraction of Matrix ElementsElements
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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
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Multiplication of Matrix Multiplication of Matrix ElementsElements
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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
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Division of Matrix Division of Matrix ElementsElements
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Powers of Matrix Powers of Matrix ElementsElements
The square sign is The square sign is enclosed in parenthesesenclosed in parentheses..
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Powers of Matrix Powers of Matrix ElementsElements
The square sign is The square sign is enclosed in parenthesesenclosed in parentheses
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Matrix Algebra Matrix Algebra OperationsOperations
on Matrices
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Addition of MatricesAddition of Matrices
3 x 1 1 x 3 3 x 3
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Major Addition of Major Addition of MatricesMatrices
1 + 1 = 21 + 2 = 31 + 3 = 4
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Major Addition of Major Addition of MatricesMatrices
2 + 1 = 32 + 2 = 42 + 3 = 5
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Major Addition of Major Addition of MatricesMatrices
3 + 1 = 43 + 2 = 53 + 3 = 6
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Minor Addition of Minor Addition of MatricesMatrices
(1+1) + (2+2) + (3+3) = 12
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Subtraction of Subtraction of MatricesMatrices
1 x 3 3 x 1 1 x 1
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Minor Subtraction of Minor Subtraction of MatricesMatrices
(1-1) + (2-2) + (3-3) =0
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Major Subtraction of Major Subtraction of MatricesMatrices
1 - 1 = 01 - 2 = -11 - 3 = -2
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Major Subtraction of Major Subtraction of MatricesMatrices
2 - 1 = 12 - 2 = 02 - 3 = -1
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Major Subtraction of Major Subtraction of MatricesMatrices
3 - 1 = 23 - 2 = 13 - 3 = 0
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Multiplication of Multiplication of MatricesMatrices
3 x 2 2 x 3 3 x 3
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Multiplication of Multiplication of MatricesMatrices
(1*7) + (2*10) =27
(1*8) + (2*11) =30
(1*9) + (2*12) =33
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Multiplication of Multiplication of MatricesMatrices
(3*8) + (4*11) = 68(3*7) + (4*10) = 61
(3*9) + (4*12) = 75
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Multiplication of Multiplication of MatricesMatrices
(5*7) + (6*10) = 95(5*8) + (6*11) = 106(5*9) + (6*12) = 117
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Matrix InversionMatrix Inversion
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Matrix InversionMatrix Inversion
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Matrix InversionMatrix Inversion
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Powers of MatricesPowers of Matrices
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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
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Elements Of Elements Of StatisticsStatistics
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Algebraic MeanAlgebraic Mean
In Summation NotationIn Summation Notation
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Summation NotationSummation Notation
M XnX
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AlgebraicAlgebraic Mean Mean
In Matrix Algebra NotationIn Matrix Algebra Notation
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Matrix Algebra NotationMatrix Algebra Notation
nXM x
'1
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Matrix MultiplicationMatrix Multiplication
35
155
54321
11111
xM
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MeanMean
35
155
54321
11111
xM
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True VarianceTrue Variance
In Summation NotationIn Summation Notation
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Summation NotationSummation Notation
2
222 )(
nXXn
x
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True VarianceTrue Variance
In Matrix Algebra NotationIn Matrix Algebra Notation
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MatrixMatrix Algebra Notation Algebra Notation
2
)2(2 1)'('1
nXX
x
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Matrix Subtraction: X – X’Matrix Subtraction: X – X’
2
)2(
2
511111
54321
54321
11111
x
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Resulting Pairwise DifferencesResulting Pairwise Differences
2
)2(
2
511111
0123410123210123210143210
11111
x
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Triangulate the MatrixTriangulate the Matrix
2
)2(
2
511111
0123400123000120000100000
11111
x
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Square the Matrix ElementsSquare the Matrix Elements
2511111
01491600149000140000100000
11111
2
x
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VarianceVariance
225502 x
Sum the squared elements
Relational space
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CovarianceCovariance
In Summation NotationIn Summation Notation
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Summation NotationSummation Notation
cov xyxyn
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CovarianceCovariance
In Matrix Algebra NotationIn Matrix Algebra Notation
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Matrix Algebra NotationMatrix Algebra Notation
C D Dn
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Obtained ScoresObtained Scores
X
2 11 25 34 43 5
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Deviation ScoresDeviation Scores
D
1 22 12 01 10 2
x y
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Matrix MultiplicationMatrix Multiplication
C D Dn
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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
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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
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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
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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
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CorrelationCorrelation
In Summation NotationIn Summation Notation
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Summation NotationSummation Notation
rz znxyx y
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CorrelationCorrelation
In Matrix Algebra NotationIn Matrix Algebra Notation
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Matrix Algebra NotationMatrix Algebra Notation
R Z Zn
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Obtained ScoresObtained Scores
X
2 11 25 34 43 5
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Standard ScoresStandard Scores
Z
. .
. .
. .
. .. .
71 141141 71141 00
71 7100 141
Zx Zy
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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
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Resulting Matrix Resulting Matrix
R
5 0 2 52 5 5 0
5
. .. .
ZxZx
ZyZy
ZxZy
ZyZx
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Correlation MatrixCorrelation Matrix
R
100 5050 100. .. .