Psychology 202bAdvanced Psychological

Statistics, II

January 18, 2011

Overview

• What will we do this semester?• Accessing the syllabus.• Review of the 202a final exam.• Introduction to matrices.

Matrices

• What is a matrix?• First, what is a vector?

– A variable for which the meaning is carried by a set of values.

– Any type of variable can be vector valued.• A matrix may be thought of as a vector of

vectors.

Notation

• It is conventional to indicate that a variable is a vector or matrix by using bold face type.– Vectors are lower case: a, b.– Matrices are upper case: A, B.

• In contrast to scalars, vector and matrix variables are not italicized.

Examples of vectors

• The set of 40 Peabody scores that we analyzed last semester could be thought of as a vector of Peabody scores.

• The Peabody and Raven score for the first subject in our data set could be thought of as a vector of scores associated with that person.

Example of a matrix

• If we combine those two, the set of 40 pairs of Raven and Peabody scores is a 40-by-2 matrix.

• What are the dimensions of this matrix?

Classifications of matrices

• Various special types of matrices exist. For example, we will consider:– Symmetric matrices– Upper and lower triangular matrices– Diagonal matrices

Symmetric matrices

• A matrix is symmetric if the elements on both sides of the diagonal that runs from the upper left corner to the lower right corner are reflections of each other:

397924741

Symmetric matrices (cont.)

• Examples of symmetric matrices that we frequently encounter in statistics include– Correlation matrices– Covariance matrices

Triangular matrices

• A triangular matrix is one that consists solely of zeroes on one side of the diagonal:

11663091100320004

Triangular matrices (cont.)

• That was a lower triangular matrix: The non-zero values were on and below the diagonal.

• An upper triangular matrix would have zeroes below the diagonal, and the non-zero values would all be on or above the diagonal.

Diagonal matrices

• A diagonal matrix is one in which all values that are not on the diagonal are zero:

300020001

Matrix multiplication

• How does matrix multiplication work?– Examples on board.– In-class exercise.

• Note that matrix multiplication is not commutative.

Matrix multiplication vocabulary

• The matrix on the left in matrix multiplication is called the “premultiplier.”

• The matrix on the right is called the “postmultiplier.”

Matrix transposition

• The transpose of a matrix is a matrix where the rows and columns are reversed.

• Example:

101102313121221

122031112132011

Matrix transposition (cont.)

• Sometimes when a multiplication problem does not conform, it will when transposed.

• Example on the board.

Next time

• Matrix division: the inverse matrix.• Manipulating matrices in R.• The relevance of matrices to regression.