1. MIT 18.06 Problem Set 9 — how this assessment works

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 9” 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 PSET9 Problem 1 — From LDL^T to the Cholesky factor (4 pts)

With positive pivots in $D$ , the factorization $A=LDL^{T}$ becomes $L\sqrt{D}\sqrt{D}L^{T}$ . (Square roots of the pivots give $D=\sqrt{D}\sqrt{D}$ .) Then $C=\sqrt{D}L^{T}$ yields the Cholesky factorization $A=C^{T}C$ which is "symmetrized LU".

From $C=[ \begin{array}{cc} 3 & 1 \\ 0 & 2 \end{array} ]$ find $A$ . From $A=[ \begin{array}{cc} 4 & 8 \\ 8 & 25 \end{array} ]$ find $C=\mathbf{chol}(A)$ .

3. MIT 18.06 PSET9 Problem 2 — Cholesky factors of two positive definite matrices (4 pts)

In the Cholesky factorization $A=C^{T}C$ , with $C^{T}=L\sqrt{D}$ , the square roots of the pivots are on the diagonal of $C$ . Find $C$ (upper triangular) for

A=[ \begin{array}{ccc} 9 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 8 \end{array} ] \quad \text{and} \quad A=[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 7 \end{array} ]

4. MIT 18.06 PSET9 Problem 3 — Completing the square and the second pivot (4 pts)

The symmetric factorization $A=LDL^{T}$ means that $\mathbf{x}^{T}A\mathbf{x}=\mathbf{x}^{T}LDL^{T}\mathbf{x}$ :

[ \begin{array}{cc} x & y \end{array} ] [ \begin{array}{cc} a & b \\ c & d \end{array} ] [ \begin{array}{c} x \\ y \end{array} ]=[ \begin{array}{cc} x & y \end{array} ] [ \begin{array}{cc} 1 & 0 \\ b/a & 1 \end{array} ] [ \begin{array}{cc} a & 0 \\ 0 & (ac-b^2)/a \end{array} ] [ \begin{array}{cc} 1 & b/a \\ 0 & 1 \end{array} ] [ \begin{array}{c} x \\ y \end{array} ]

The left side is $ax^2+2bxy+cy^2$ . The right side is $a(x+\frac{b}{a}y)^2+\underline{\qquad} y^2$ . The second pivot completes the square! Test with $a=2, b=4, c=10$ .

5. MIT 18.06 PSET9 Problem 4 — Hessians: a minimum and a saddle point (4 pts)

For $F_1(x, y)=x^4/4+x^2+x^2y+y^2$ and $F_2(x, y)=x^3+xy-x$ , find the second-derivative matrices $H_1$ and $H_2$ :

H=(\begin{array}{cc} \frac{\partial^2 F}{\partial x^2} & \frac{\partial^2 F}{\partial x \partial y} \\ \frac{\partial^2 F}{\partial y \partial x} & \frac{\partial^2 F}{\partial y^2} \end{array}).

$H_1$ is positive-definite so $F_1$ is concave up (=convex). Find the minimum point of $F_1$ and the saddle point of $F_2$ (look only where the first derivatives are zero).

6. MIT 18.06 PSET9 Problem 5 — Groups among families of matrices (12 pts)

A group of nonsingular matrices includes $AB$ and $A^{-1}$ if it includes $A$ and $B$ . "Products and inverses stay in the group." Which of these are groups (as in 2.7.37)? Invent a "subgroup" of two of these groups (not $I$ by itself=the smallest group).

(a) Positive definite symmetric matrices $A$ .
(b) Orthogonal matrices $Q$ .
(c) All exponentials $e^{tA}$ of a fixed matrix $A$ .
(d) Matrices $P$ with positive eigenvalues.
(e) Matrices $D$ with determinant 1.

7. MIT 18.06 PSET9 Problem 6 — Positive eigenvalues of AB for positive definite A and B (12 pts)

When $A$ and $B$ are symmetric positive definite, $AB$ might not even be symmetric. But its eigenvalues are still positive. Start from $AB\mathbf{x}=\lambda\mathbf{x}$ and take dot products with $B\mathbf{x}$ . Then prove $\lambda > 0$ .

8. MIT 18.06 PSET9 Problem 7 — The 5-by-5 sine matrix and the eigenvalues of K (12 pts)

Write down the 5 by 5 sine matrix $S$ from Worked Example 6.5 D, containing the eigenvectors of $K$ when $n=5$ and $h=1/6$ . Multiply $K$ times $S$ to see the five positive eigenvalues.

Their sum should equal the trace 10. Their product should be $\det K=6$ .

9. MIT 18.06 PSET9 Problem 8 — A^T C A is positive definite (12 pts)

If $A$ has full column rank, and $C$ is positive-definite, show that $A^{T}CA$ is positive definite.

10. MIT 18.06 PSET9 Problem 9 — Free-free chain: balanced forces and all solutions (4 pts)

In the free-free case when $A^{T}CA$ in equation (9) is singular, add the three equations $A^{T}CA\mathbf{u}=\mathbf{f}$ to show that we need $f_1+f_2+f_3=0$ . Find a solution to $A^{T}CA\mathbf{u}=\mathbf{f}$ when the forces $\mathbf{f}=(-1, 0, 1)$ balance themselves. Find all solutions!

11. MIT 18.06 PSET9 Problem 10 — The fixed-free differential equation -y'(x) = f(x) (4 pts)

In the fixed-free problem, the matrix $A$ is square and invertible. We can solve $A^{T}\mathbf{y}=\mathbf{f}$ separately from $A\mathbf{u}=\mathbf{e}$ . Do the same for the differential equation:

\text{Solve} \quad-\frac{dy}{dx}=f(x) \quad \text{with} \quad y(1)=0. \quad \text{Graph} \quad y(x) \quad \text{if} \quad f(x)=1.

12. MIT 18.06 PSET9 Problem 11 — Five springs, four masses, both ends fixed (4 pts)

For five springs and four masses with both ends fixed, what are the matrices $A$ and $C$ and $K$ ? With $C=I$ solve $K\mathbf{u}=\text{ones}(4)$ .

13. MIT 18.06 PSET9 Problem 12 — MATLAB: 100 masses, fixed-fixed and fixed-free (12 pts)

(MATLAB) Find the displacements $u(1), \ldots, u(100)$ of 100 masses connected by springs all with $c=1$ . Each force is $f(i)=0.01$ . Print graphs of $\mathbf{u}$ with fixed-fixed and fixed-free ends. Note that $\text{diag}(\text{ones}(n, 1), d)$ is a matrix with $n$ ones along the diagonal $d$ . This print command will graph a vector $u$ :

\begin{array}{l}\text{plot(u, '+'); xlabel('mass number'); ylabel('movement'); print}\end{array}

14. MIT 18.06 PSET9 Problem 13 — MATLAB: forward, centered and backward differences (12 pts)

(MATLAB) Chemical engineering has a first derivative of $du/dx$ from fluid velocity as well as $d^2u/dx^2$ from diffusion. Replace $du/dx$ by a forward difference, then a centered difference, then a backward difference, with $\nabla x=\frac{1}{8}$ . Graph your numerical solutions of

-\frac{d^2u}{dx^2}+10\frac{du}{dx}=1 \text{ with } u(0)=u(1)=0.