1. MIT 18.06 Problem Set 10 — how this assessment works

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 10” 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 13 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.

2. MIT 18.06 PSET10 Problem 1 — Jordan blocks of different sizes are not similar (4 pts)

These Jordan matrices have eigenvalues $0, 0, 0, 0$ . They have two eigenvectors (one from each block). But the block sizes don't match and they are not similar:

J=(\begin{array}{cc|cc} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}) \qquad \text{and} \qquad K=(\begin{array}{ccc|c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 \end{array})

For any matrix $M$ , compare $JM$ with $MK$ . If they are equal show that $M$ is not invertible. Then $M^{-1}JM=K$ is impossible; $J$ is not similar to $K$ .

3. MIT 18.06 PSET10 Problem 2 — A transpose is always similar to A (4 pts)

Prove that $A^T$ is always similar to $A$ (we know that the $\lambda$ 's are the same):

For one Jordan block $J_i$ : find $M_i$ so that $M_i^{-1}J_iM_i=J_i^T$ (see example 3).
For any $J$ with blocks $J_i$ : build $M_0$ from blocks so that $M_0^{-1}JM_0=J^T$ .
For any $A=MJM^{-1}$ : Show that $A^T$ is similar to $J^T$ and so to $J$ and so to $A$ .

4. MIT 18.06 PSET10 Problem 3 — Five true statements about similarity (4 pts)

Why are these statements all true?

(a) If $A$ is similar to $B$ then $A^2$ is similar to $B^2$ .

(b) $A^2$ and $B^2$ can be similar when $A$ and $B$ are not similar.

(c) $\begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix}$ is similar to $\begin{pmatrix} 3 & 1 \\ 0 & 4 \end{pmatrix}$ .

(d) $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ is not similar to $\begin{pmatrix} 3 & 1 \\ 0 & 3 \end{pmatrix}$ .

(e) If we exchange rows 1 and 2 of $A$ , and then exchange columns 1 and 2 the eigenvalues stay the same. In this case $M=?$

5. MIT 18.06 PSET10 Problem 4 — All eigenvalues zero implies A^n is the zero matrix (12 pts)

If an $n \times n$ matrix $A$ has all eigenvalues $\lambda=0$ prove that $A^n$ is the zero matrix.

6. MIT 18.06 PSET10 Problem 5 — Shifted QR: A_2 is similar to A_1 (12 pts)

For the shifted $QR$ method in the Worked Example 6.6 B, show that $A_2$ is similar to $A_1$ . No change in eigenvalues, and the $A$ 's quickly approach a diagonal matrix.

7. MIT 18.06 PSET10 Problem 6 — Similar to its own inverse (12 pts)

If $A$ is similar to $A^{-1}$ , must all the eigenvalues equal $1$ or $-1$ ?

8. MIT 18.06 PSET10 Problem 7 — SVD of the Fibonacci matrix (4 pts)

Find the eigenvalues and unit eigenvectors of $A^T A$ and $AA^T$ . Keep each $Av=\sigma u$ for the Fibonacci matrix $A=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ . Construct the singular value decomposition and verify that $A$ equal $sU\Sigma V^T$ .

9. MIT 18.06 PSET10 Problem 8 — SVD when the columns are orthogonal (4 pts)

Suppose $A$ has orthogonal columns $w_1, \ldots, w_n$ of lengths $\sigma_1, \ldots, \sigma_n$ . What are $U, \Sigma$ and $V$ in the SVD?

10. MIT 18.06 PSET10 Problem 9 — SVD of the 4 by 3 first difference matrix (12 pts)

The $1,-1$ first difference matrix $A$ has $A^T A$ the second difference matrix. The singular vectors of $A$ are sine vectors $V$ and cosine vectors $u$ . Then $Av=\sigma u$ is the discrete form of $d/dx(\sin cx)=c(\cos cx)$ . This is the best SVD I have seen.

A=(\begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 &-1 & 1 \\ 0 & 0 &-1 \end{array}) \qquad A^T A=(\begin{array}{ccc} 2 &-1 & 0 \\ -1 & 2 &-1 \\ 0 &-1 & 2 \end{array}).

Then the orthogonal sine matrix is

V=\frac{1}{\sqrt{2}} (\begin{array}{ccc} \sin \pi/4 & \sin 2\pi/4 & \sin 3\pi/4 \\ \sin 2\pi/4 & \sin 4\pi/4 & \sin 6\pi/4 \\ \sin 3\pi/4 & \sin 6\pi/4 & \sin 9\pi/4 \end{array}).

(a) Put numbers in $V$ : The unit eigenvectors of $A^T A$ are singular vectors of $A$ . Show that the columns of $V$ have $A^T Av=\lambda v$ with $\lambda=2-\sqrt{2}, 2, 2+\sqrt{2}$ .

(b) Multiply $AV$ and verify taht its columns are orthogonal. They are $\sigma_1u_1$ and $\sigma_2u_2$ and $\sigma_3u_3$ . The first columns of the cosine matrix $U$ are $u_1, u_2, u_3$ .

(c) Since $A$ is $4 \times 3$ weneed a fourth orthogonal vector $u_4$ . It comes from the nullspace of $A^T$ . What is $u_4$ ?

11. MIT 18.06 PSET10 Problem 10 — The fourth Legendre polynomial (4 pts)

The first three Legendre polynomials are $1, x, x^2-1/3$ . Choose $c$ so that the fourth polynomial $x^3-cx$ is orthogonal to the first three. All integrals go from $-1$ to $1$ .

12. MIT 18.06 PSET10 Problem 11 — Square wave: three Fourier integrals (4 pts)

For the square wave $f(x)$ in Example 3 show that

\int_0^{2\pi} f(x) \cos x \, dx=0 \qquad \int_0^{2\pi} f(x) \sin x \, dx=4 \qquad \int_0^{2\pi} f(x) \sin 2x \, dx=0.

Which three Fourier coefficients come from those integrals?

13. MIT 18.06 PSET10 Problem 12 — The 5 by 5 differentiation matrix (12 pts)

The functions $1, \cos x, \sin x, \cos 2x, \sin 2x, \ldots$ are a basis for a Hilberts space. Write the derivatives of those first five functions as combinations of the same five functions. What is the $5 \times 5$ “differentiation matrix” for those functions?

14. MIT 18.06 PSET10 Problem 13 — Square pulse approaching a delta function (12 pts)

Find the Fourier coefficients $a_k$ and $b_k$ of the square pulse $F(x)$ centered at $x=0$ : $f(x)=1/h$ for $|x| \le h/2$ and $F(x)=0$ for $h/2 < |x| \le \pi$ . As $h \to 0$ , this $F(x)$ approaches a delta function. Find the limits of $a_k$ and $b_k$ .