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: Diagonalization Suppose we are given the following matrix ```{r} A <- matrix(seq(3, 27, by=3), nrow=3, ncol=3) ``` #### Question 1: Compute the rank of the matrix `A` . Then compute the eigen values of the matrix `A`. How is the rank related to the eigenvalues of the matrix? #### Question 2: Use the eigen decomposition as follows: extract the matrix of the eigenvectors by calling the variable `$vectors` and assign these vectors to a matrix called `C`. Check if it is possible to invert this matrix and if so, define the inverse `C_inv`. Explain why this is possible or why not. #### Question 3: Now try the following, multiply `A` by `C_inv` on the left and `C` on the right. What do you notice about the values off diagonal? Try using the following comparison to see how close the elements are to the value zero, ```{r, eval=FALSE} abs(product) <= ones(3) * 10e-14 ``` Take the diagonal elements of the product above and find the absolute difference of these from the eigenvalues of `A`. What do you notice about the result.