1. Psychology 202b Advanced Psychological Statistics, II January 18, 2011

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

3. 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.

4. 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.

5. 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.

6. 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?this matrix

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

8. 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:

9. Symmetric matrices (cont.) Examples of symmetric matrices that we frequently encounter in statistics include –Correlation matrices –Covariance matrices

10. Triangular matrices A triangular matrix is one that consists solely of zeroes on one side of the diagonal:

11. 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.

12. Diagonal matrices A diagonal matrix is one in which all values that are not on the diagonal are zero:

13. Matrix multiplication How does matrix multiplication work? –Examples on board. –In-class exercise. Note that matrix multiplication is not commutative.

14. Matrix multiplication vocabulary The matrix on the left in matrix multiplication is called the “premultiplier.” The matrix on the right is called the “postmultiplier.”

15. Matrix transposition The transpose of a matrix is a matrix where the rows and columns are reversed. Example:

16. Matrix transposition (cont.) Sometimes when a multiplication problem does not conform, it will when transposed. Example on the board.

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