Notation Fundamentals
Definition. A matrix is defined as an ordered array
of numbers, of dimensions p, q . Our standard
notation for a matrix A of order p, q will be:
p Aq
(1)
There are numerous other notations. For example,
one might indicate a matrix of
The Laws of Linear Combination
James H. Steiger
Goals for this Module
In this module, we cover
What is a linear combination? Basic definitions and
terminology
Key aspects of the behavior of linear combinations
The Mean of a linear combination
The Variance
Dening a Matrix in R
Extracting Pieces of a Matrix
Combining Matrices
Basic Matrix Operations
Matrix Algebra in R A Minimal Introduction
James H. Steiger
Department of Psychology and Human Development
Vanderbilt University
Multilevel Regression Modeling,
Inverse of a Square Matrix
For an N N square matrix A, the inverse of A,
A 1, exists if and only if A is of full rank, i.e., if
and only if no column of A is a linear combination
of the others. A 1 is the unique matrix that satisfies
A 1A = AA 1 = I
(1)
C
Random Vectors and Random Matrices
In this section, we extend our results on linear
combinations of variables to random vector
notation. The generalization is straightforward.
Definition (Random Vector). A random vector
is a vector whose elements are ran
Eigenvalues, Eigenvectors, Matrix
Factoring, and Principal Components
The eigenvalues and eigenvectors of a square matrix play
a key role in some important operations in statistics. In
particular, they are intimately connected with the
determination of th
Introduction
Expected Value of a Random Variable
Variance of a Random Variable
Covariance of Two Random Variables
Correlation of Two Random Variables
Algebra of Variances and Covariances
Joint Distributions and Conditional Expectation
Matrix Expected Valu
Introduction to R
Load the R package by double-clicking the R icon.
After some preliminary messages, youll see the prompt.
>
To see some demos, type
> demo()
A demo window will open up
Click back on the console window, and type
>demo(graphics)
You may nee
Introduction
The SVD
An Example
Non-Uniqueness of Eigenvectors
The Great SVD Mystery
James H. Steiger
Department of Psychology and Human Development
Vanderbilt University
P312, 2012
James H. Steiger
The Great SVD Mystery
Introduction
The SVD
An Example
No
The Multiple Regression Model
Some Key Regression Terminology
The Kids Data Example
Understanding Regression Coecients
Statistical Testing in the Fixed Regressor Model
Variable Selection in Multiple Regression
Variable Selection in R
Information-Based Sel
Matrix Algebra of Some Sample Statistics
Variance of a Linear Combination
Variance-Covariance Matrix of Several Linear Combinations
Covariance Matrix of Two Sets of Linear Combinations
Matrix Algebra of Sample Statistics
James H. Steiger
Department of Psy
Correlation and Covariance
James H. Steiger
Goals for Today
Introduce the statistical concepts of
Covariance
Correlation
Investigate invariance properties
Develop computational formulas
Covariance
So far, we have been analyzing summary
statistics that de
Regression Analysis
Some Examples
Revisiting Basic Regression Results
Anscombes Quartet
Smoothing the Mean Function
The Scatterplot Matrix
Two Bivariate Regression Models
Where from Here?
Introduction to Regression and Correlation
James H. Steiger
Departm
Matrix Multiplication
Matrix multiplication is an operation with
properties quite different from its scalar
counterpart.
To begin with, order matters in matrix
multiplication. That is, the matrix product AB need
not be the same as the matrix product BA. I
Why Matrix Algebra?
Basic Denitions
Matrix Operations
Matrix Algebra A Minimal Introduction
James H. Steiger
Department of Psychology and Human Development
Vanderbilt University
Multivariate Analysis, 2012
P312
Matrix Algebra
Why Matrix Algebra?
Basic Den