Homework 4

This HW is due on 18th November 2024. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 18th November 2024. Please submit the HW as a single pdf file - you can scan / take a photo of your completed HW (handwritten) and convert it into a pdf file (please name your file first.last_hw4.pdf). To get credit, please provide all details and give complete reasoning for all your work.

Exercise 1 (2 Points) Find the eigenvalues of the matrix A given below, alongwith the algebraic multiplicties of each eigenvalue A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & -2 \\ 3 & 6 & -5 \end{bmatrix}

Exercise 2 (2 Points) Do the eigenvectors of the matrix B given below form a basis of \mathbb{R}^3? B= \begin{bmatrix} 2 & 0 & 0 \\ 1 & -1 & 1 \\ 2 & 1 & -1 \end{bmatrix}

Exercise 3 (2 Points) Find \underline{\textbf{all}} the 2 \times 2 matrices for which 7 is an eigenvalue with the corresponding eigenspace being the entire \mathbb{R}^2.

Exercise 4 (2 Points) Let C = \begin{bmatrix} 0& 2 & 0 & 0 \\ m & 0 & 2 & 0 \\ 0 & m & 0 & 2 \\ 0 & 0 & m & 0 \end{bmatrix}

Is there a real value of m such that matrix C is diagonalizable ?

Exercise 5 (2 Points) For the matrix D given below, find a singular value decomposition in the form D= \sigma_1 \mathbf{u}_1 \mathbf{v}_1^T + \sigma_2 \mathbf{u}_2 \mathbf{v}_2^T \begin{bmatrix} 6 & 2 \\ -7 & 6 \end{bmatrix}