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. .. .