Write your own function $\operatorname{sim}(A, C)$ for performing a similarity transformation $B=C^{-1} A C$. The function should be on your own disc, call matrices $A$ and $C$ from the matrix command window, and return $B$ to the command window. Check your function out on the similarity transformations of just the previous exercise.

Use cond $(A)$ to determine the condition number of the given matrix $A$. Confirm that it does indeed equal the ratio of the largest singular value to the smallest.

$$A=\left[\begin{array}{rrrr} -2 & 7 & 1 & 3 \\ 4 & 0 & -5 & 6 \\ 2 & 1 & -3 & 1 \\ 4 & 5 & 2 & -8 \end{array}\right] \\$$

Use circles(A) to sketch circles having centers of the origin and radii $1,2 ,$ and $3$ in the spaces with inner products defined by the following matrices.

(a)

$$A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$$

(b)

$$B=\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]$$

(c)

$$C=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$$

Determine the equations of these circles.

Determine the least squares solution to the following systems.

(a)

$$\begin{array}{r} x+y=3 \\ x-y=1 \\ 2 x+3 y=5 \end{array}$$

(b)

$$\begin{array}{r} 2 x-y=0 \\ x+2 y=3 \\ x+y=1 \end{array}$$

(c)

$$\begin{aligned} 3 x+2 y &=5 \\ x-y &=1 \\ 2 x+y &=4 \end{aligned}$$

Use SVD to determine the rank of the matrix

$$A=\left[\begin{array}{rrrr}2 & -1 & 3 & 1 \\ 1 & 4 & -2 & 3 \\ 5 & 2 & 4 & 5 \\ 7 & 10 & 0 & 11\end{array}\right]$$

. Display $S$ to sixteen decimal places using the command format long.

Use MATLAB to perform the similarity transformation $B=C^{-1} A C$, where

(a)

$$A=\left[\begin{array}{rr}7 & -10 \\ 3 & -4\end{array}\right]$$

and

$$C=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]$$

(b)

$$A=\left[\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right]$$

and

$$C=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]$$

(c)

$$A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 1 & 0\end{array}\right]$$

and

$$C=\left[\begin{array}{rrr}3 & 5 & -1 \\ -2 & -3 & 1 \\ -1 & -2 & 1\end{array}\right]$$

(d)

$$A=\left[\begin{array}{rrr}5 & 2 & 3 \\ 8 & 2 & 1 \\ -3 & 2 & 5\end{array}\right]$$

and

$$C=\left[\begin{array}{rrr}7 & -2 & 3 \\ 2 & 0 & 1 \\ 9 & -3 & 5\end{array}\right]$$

Use MATLAB to find the most and least efficient ways of performing the following products. $A$ is $2 \times 45, B$ is $45 \times 45$, and $C$ is $45 \times 3$.

(a) $A B C$ (b) $A B^2$ (c) $A B^3$ (d) $A B^2 C$

Determine the equation of the polynomial of degree three through the points $(1,7),(2,3),(3,9),(4,21)$. Compute the condition number of the matrix of coefficients in the system of equations that arises.

Find the matrix that defines a reflection in the $y$-axis. Determine and sketch the image of the unit square under this transformation.

An economy of two industries has the following I/O matrix $A$ and demand matrix $D$. Apply the Leontief model to this situation. Why does it not work? Explain the situation from both a

$$A=\left[\begin{array}{ll} 0.8 & 0.4 \\ 0.2 & 0.6 \end{array}\right], D=\left[\begin{array}{l} 20 \\ 32 \end{array}\right]$$

mathematical and practical viewpoint. Extend your results to the general model.

Use the computer to find the rank of the matrix

$$\left[\begin{array}{rrrrr} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}\right]$$

(a) Explain the result. (b) Generalize the result.

Solve the given systems using left division. Check your answers using rref.

$$\begin{aligned} &x+2 y-z=2 \\ &x+y+2 z=0 \\ &x-y-z=1 \end{aligned}$$

Use MATLAB to investigate the given matrix $A$. Determine as many properties as you can. (Hint: Look at the vectors that make up the rows of $A, \operatorname{det}(\mathrm{A})$, etc.) Construct two other matrices, one $2 \times 2$, the other $3 \times 3$, that have the same properties as $A$.

$$A=\left[\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array}\right]$$

Consider

$$P=\left[\begin{array}{rrr}0 & -2 & 3 \\ 1 & 4 & 5\end{array}\right]$$

and

$$Q=\left[\begin{array}{rrr}5 & 2 & -1 \\ 0 & 6 & 3\end{array}\right]$$

.

Compute $4 p_{13}-5 q_{23}+2\left(p_{22}\right)^2$.