Instructor: Gustavo de los Campos ( gustavoc@msu.edu )
Description. In this independent study we will reivew basic matrix algebra and computational methods with a focus on biostatistical applications.
Textbook: Matrix Algebra Useful for Statistics by Shayle R. Searle.
Content. We will cover chapters 1-11, 14 and 15 of the textbook, plus a few additional topics.
Approach. Students will read one chapter per week and complete 2-3 problems (mostly from the book). We will meet once a week for half an hour to discuss the chapter assigned to the week. Every week one student will lead the discussion, this will include presenting a summary of the chapter, grading the HW and leading the discussion.
Requirements: (i) Read the chapter assigned. (ii) Complete and turn in the HW assigned (usually 2 problems from the chapter). (iii) Lead discussions.
Day/Time. We will meet every Monday at 3:00-3:30pm in room 637 (Fee Hall, 909 Fee Road, 6th floor, Room 637).
Evaluation will we be based on: (i) attendance, (ii) HW, and (iii) discussions lead.
Chapter 1 & Chapter to up to page 32 (02/01/2016)
- Probelms: Ch.2 problem 3 (solve the problem using R code); Chapter 2 problem 11; chapter 2, problem 6.
- For discussion: dot notation and colSums/rowSums operators in R; Comprehensions in Julia.
Chapter 2 (pages 32-53, 02/01/2016)
Chapter 3 (Special Matrices, pages 60-83)
Homework
- show that (AB)'=B'A'
- Express colSums, colMeans, rowSums and rowMeans using matrix algebra
- Express scale(X,center=T,scale=F) using matrix algebra
- Show that if x'Ax is positive definite, then x'(A+I)x is also positive definite.
Chapter 4 (Determinants, 2/17/2016)
- Additional read: https://en.wikipedia.org/wiki/Determinant
- HW: list all the properties that you know about determinants (use both the book and the above link).
- Examples of properties are:
- det(AB)=? if A is diagonal, '
- det(A)=?,
- if A is lower-triangular, det(A)=?
- if A is singlular, det(A)=?
- if A'A=I, det(A)=?....
Chapter 5 (Determinants)
Topics to be discussed:
- Use of inverses (focus on solving systems of linear equations)
- Definition of inverse
- Relationship to determinants
- Properties (page 130)
- Inverses of positive definite matrices using cholesky and eigen decompositions.
Chapter 6 (Rank, 02/29/2016)
HW:
- solve problems: 1, 8 and 10.
- Discuss relationships between eigenvalues of symmetric matrices and rank, discuss relationship between positive definitness and rank.
Chapter 7 (Person in charge TBA, 03/07/2016)
Chapter (Person in charge TBA, 03/14/2016)
Chapter (Person in charge TBA, 03/21/2016)
Chapter (Person in charge TBA, 03/28/2016)
Chapter (Person in charge TBA, 04/04/2016)
Special topics: Krnoecker Products (Person in charge TBA, 04/11/2016)
Special topics: Multivariate Normal Distribution (Person in charge TBA, 04/18/2016)