PSY544 – Introduction to Factor Analysis
Homework assignment 1, Fall 2018
Due midnight, October 17, 2018
Suppose we have the following matrices:
𝑨 = [
8 4
3 12
] 𝑩 = [
6 3 9
11 0 1
] 𝑪 = [
0 −8 0
0 0 3
5 −7 0
]
𝑫 = [
13 5
1 0
7 3
] 𝑿 =
[
8
1
2
5
4]
Compute the following. Show your work.
1. X’X
2. DA
3. CB
4. (DB)C
5. BD + A
6. C + D
7. Is B’A equal to AB?
8. Compute |A|. We haven’t covered this in class, so look up online how to do it by hand.
Now, suppose we have a simple, ordinary linear regression model based on N = 5 observations.
Using scalar notation, we would write the model as follows:
𝑦𝑖 = 𝛽0 + 𝛽1 𝑥𝑖 + 𝜀𝑖
for i = 1, ..., N. The 𝑥𝑖s are the predictor scores, and the 𝜀𝑖s are error terms that have zero
means (by definition). Therefore, the expected value of 𝑦𝑖 is a linear function of the regression
coefficients and the predictor scores:
𝐸(𝑦𝑖) = 𝜇𝑖 = 𝛽0 + 𝛽1 𝑥𝑖
Let the N x 1 vector of means be denoted as 𝝁. Write down each element of this vector in terms
of the regression coefficients and predictor scores:
𝝁 =
[
𝜇1
𝜇2
𝜇3
𝜇4
𝜇5]
=
[
𝛽0 + 𝛽1 𝑥1
]
Given what you’ve just written down, re-express the mean vector 𝝁 as a matrix product
between a N x 2 matrix X and a 2 x 1 vector 𝜷:
𝝁 =
[
𝜇1
𝜇2
𝜇3
𝜇4
𝜇5]
= 𝑿𝜷 =
[
1 𝑥1
]
[ ]
Now, let’s call the N x 1 vector from the previous equation y. Also, let’s aggregate all the 𝜀𝑖s
into an N x 1 vector of error terms. How would you formulate the original regression equation
using matrix notation?
𝒚 =
[
1 𝑥1
]
[ ] +
[ ]
= _____ + 𝜺