David J' Krus presents Matrix Algebra for Social Sciences

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David J. KruspresentsMatrix Algebrafor
Social Sciences
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Introduction toMatrix Algebra
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Dimensions of a Matrix

• Number of Rows 2
• Number of Columns3
• A 2 x 3 Matrix

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Elements of a Matrix
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Principal Diagonal Elements
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Off-Diagonal Elements
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Nomenclature of Matrices

• Rectangular
• Square
• Symmetric
• Skew Symmetric

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Transpose
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Triangulation
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Matrix Algebra Operations
on Matrix Elements
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Addition of Matrix Elements
2 x 2
2 x 2
2 x 2

• All matrices must have the the same dimensions.
• The plus sign is enclosed in parentheses.

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Addition of Matrix Elements
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Subtraction of Matrix Elements
2 x 2
2 x 2
2 x 2

• All matrices must have the the same dimensions.
• The subtraction sign is enclosed in parentheses.

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Subtraction of Matrix Elements
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Multiplication of Matrix Elements
2 x 2
2 x 2
2 x 2

• All matrices must have the the same dimensions.
• The multiplication sign is enclosed in
  parentheses.

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Multiplication of Matrix Elements
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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.

2 x 2
2 x 2
2 x 2
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Division of Matrix Elements
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Powers of Matrix Elements

• The square sign is enclosed in parentheses.

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Powers of Matrix Elements

• The square sign is enclosed in parentheses

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Matrix Algebra Operations
on Matrices
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Addition of Matrices
3 x 1
1 x 3
3 x 3
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Major Addition of Matrices
1 1 2
1 2 3
1 3 4
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Major Addition of Matrices
2 1 3
2 2 4
2 3 5
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Major Addition of Matrices
3 1 4
3 2 5
3 3 6
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Minor Addition of Matrices
(11) (22) (33) 12
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Subtraction of Matrices
1 x 3
3 x 1
1 x 1
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Minor Subtraction of Matrices
(1-1) (2-2) (3-3) 0
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Major Subtraction of Matrices
1 - 1 0
1 - 2 -1
1 - 3 -2
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Major Subtraction of Matrices
2 - 1 1
2 - 2 0
2 - 3 -1
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Major Subtraction of Matrices
3 - 1 2
3 - 2 1
3 - 3 0
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Multiplication of Matrices
3 x 2
2 x 3
3 x 3
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Multiplication of Matrices
(17) (210) 27
(18) (211) 30
(19) (212) 33
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Multiplication of Matrices
(37) (410) 61
(38) (411) 68
(39) (412) 75
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Multiplication of Matrices
(57) (610) 95
(58) (611) 106
(59) (612) 117
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Matrix Inversion
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Matrix Inversion
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Matrix Inversion
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Powers of Matrices
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Powers of Matrices
(11) (23) 7
(12) (24) 10
(31) (43) 15
(32) (44) 22
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Elements Of Statistics
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Algebraic Mean

• In Summation Notation

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Summation Notation
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Algebraic Mean

• In Matrix Algebra Notation

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Matrix Algebra Notation
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Matrix Multiplication
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Mean
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True Variance

• In Summation Notation

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Summation Notation
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True Variance

• In Matrix Algebra Notation

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Matrix Algebra Notation
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Matrix Subtraction X X
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Resulting Pairwise Differences
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Triangulate the Matrix
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Square the Matrix Elements
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Variance
Sum the squared elements
Relational space
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Covariance

• In Summation Notation

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Summation Notation
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Covariance

• In Matrix Algebra Notation

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Matrix Algebra Notation
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Obtained Scores
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Deviation Scores
x
y
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Matrix Multiplication
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Matrix Multiplication
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Diagonal ElementsSums of Squares
xx
yy
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Off-Diagonal ElementsCross-Products
xy
yx
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Variance-Covariance Matrix
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Correlation

• In Summation Notation

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Summation Notation
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Correlation

• In Matrix Algebra Notation

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Matrix Algebra Notation
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Obtained Scores
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Standard Scores
Zx
Zy
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Matrix Multiplication ZZ
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Resulting Matrix
ZxZx
ZxZy
ZyZy
ZyZx
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Correlation Matrix