# Activity 10/20/2021 ## STAT 445 / 645 -- Section 1001
Instructor: Colin Grudzien
## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ## Activities: To perform some matrix operations, it can be helpful to use some non-base functions designed for this. We will require the library matlab {r} require("matlab")  ### Activity 1: Linear inverse problem #### Question 1: Suppose the matrix A from the last activity {r} A <- matrix(seq(3, 27, by=3), nrow=3, ncol=3)  defines a linear inverse problem for an unknown vector $\mathbf{x}$ and an observed vector, {r} b <- ones(3,1)  Is there any vector $\mathbf{x}$ that satisfies, $$\mathbf{A} \mathbf{x} = \mathbf{b}?$$ Why does such a choice exist or why not? #### Question 2: {r} b <- zeros(3,1) b  Is there any vector $\mathbf{x}$ that satisfies, $$\mathbf{A} \mathbf{x} = \mathbf{b}?$$ Why does such a choice exist or why not? #### Question 3: Set a random seed to be equal to zero and generate a 4-by-4 dimension matrix with entries given by a sample size $n=16$ from the standard normal. Suppose that this defines the relationship between the observed variable {r} b <- ones(4,1)  and an unknown vector $\mathbf{x}$ as, $$\mathbf{A}\mathbf{x}= \mathbf{b}.$$ Does any x as above exist? If one does find the solution. Explain why the solution exists or not due to the eigenvalues of the matrix. ### Activity 2: matrix norms Recall that matrix norms can be computed in R with the norm function. In the following we will consider some of their properties. #### Question 1: Compute the eigenvalues of the matrix B computed from matrix A below {r} B <- t(A) %*% A  then compute the sum of the square values. #### Question 2: Using the norm function, compute the norm-squared of the matrix B. Try changing the default norm -- which choice of norm-squared looks familiar?