MIT 18.06 Exam 3

This lesson reproduces the Exam “MIT 18.06 Exam 3” from MIT OpenCourseWare 18.06 Linear Algebra (Spring 2010) (https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/), licensed under CC BY-NC-SA 4.0. Work through 3 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

(40 points) Suppose $u$ is a unit vector in $R^n$, so $u^T u=1$. This problem is about the $n$ by $n$ symmetric matrix $H=I-2u u^T$.

(a) Show directly that $H^2=I$. Since $H=H^T$, we now know that $H$ is not only symmetric but also $\underline{\hspace{2cm}}$.

(b) One eigenvector of $H$ is $u$ itself. Find the corresponding eigenvalue.

(c) If $v$ is any vector perpendicular to $u$, show that $v$ is an eigenvector of $H$ and \textbf{find the eigenvalue}. With all these eigenvectors $v$, that eigenvalue must be repeated how many times? Is $H$ diagonalizable? Why or why not?

(d) Find the diagonal entries $H_{11}$ and $H_{ii}$ in terms of $u_1, \ldots, u_n$. Add up $H_{11}+\ldots+H_{nn}$ and separately add up the eigenvalues of $H$.

(30 points) Suppose $A$ is a positive definite symmetric $n$ by $n$ matrix.

(a) How do you know that $A^{-1}$ is also positive definite? (We know $A^{-1}$ is symmetric. I just had an e-mail from the International Monetary Fund with this question.)

(b) Suppose $Q$ is any \textbf{orthogonal} $n$ by $n$ matrix. How do you know that $QAQ^T=QAQ^{-1}$ is positive definite? Write down which test you are using.

(c) Show that the block matrix

is positive \textbf{semidefinite}. How do you know $B$ is not positive definite?

(30 points) This question is about the matrix

(a) Find its eigenvalues and eigenvectors.

Write the vector $u(0)=[ \begin{array}{c} 2 \\ 0 \end{array} ]$ as a combination of those eigenvectors.

(b) Solve the equation $\frac{du}{dt}=Au$ starting with the same vector $u(0)$ at time $t=0$.

In other words: the solution $u(t)$ is what combination of the eigenvectors of $A$?

(c) Find the 3 matrices in the Singular Value Decomposition $A=U \Sigma V^T$ in two steps.

–First, compute $V$ and $\Sigma$ using the matrix $A^T A$.

–Second, find the (orthonormal) columns of $U$.