# Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

Question
Matrices
Let $$U=\begin{bmatrix}\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\-\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\0&-\frac{\sqrt3}{3} \end{bmatrix} \text{ and } x=\begin{bmatrix}1 \\2 \end{bmatrix}$$
b)Compute $$\| Ux \|$$ (it should be the same as $$\| x \|$$)

2021-02-27
Step 1
Given matrices are,
$$U=\begin{bmatrix}\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\-\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\0&-\frac{\sqrt3}{3} \end{bmatrix} \text{ and } x=\begin{bmatrix}1 \\2 \end{bmatrix}$$
Step 2 Now the multiplication of the matrices is,
$$Ux=\begin{bmatrix}\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\-\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\0&-\frac{\sqrt3}{3} \end{bmatrix}\begin{bmatrix}1 \\2 \end{bmatrix}=\begin{bmatrix}\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{2\sqrt3}{3} \end{bmatrix}$$
Step 3
Norm of the matrix is,
\|Ux\|=\begin{Vmatrix}\begin{bmatrix}\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{2\sqrt3}{3} \end{bmatrix}\end{Vmatrix}=\sqrt{\left(\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3}\right)^2+\left(-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3}\right)^2+\left(-\frac{2\sqrt3}{3}\right)^2}\)
=\sqrt{\frac{2}{4}+\frac{12}{9}+2\frac{\sqrt2}{2}\frac{2\sqrt3}{2}+\frac{2}{4}+\frac{12}{9}-2\frac{\sqrt2}{2}\frac{2\sqrt3}{2}+\frac{12}{9}}\)
=\sqrt{\frac{1}{2}+\frac{4}{3}+\frac{1}{2}+\frac{4}{3}+\frac{4}{3}}=\sqrt{1+\frac{12}{3}}=\sqrt{5}\)
Step 4
Therefore, norm of the matrix is,
$$\|Ux\|=\sqrt{5}$$

### Relevant Questions

Let $$u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}$$ It can be shown that $$4u-3v-w=0$$. Use this fact (and no row operations) to find a solution to the system $$4u-3v-w=0$$ , where
$$A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}$$

Solve for X in the equation, given
$$3X + 2A = B$$
$$A=\begin{bmatrix}-4 & 0 \\1 & -5\\-3&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ -2 & 1 \\ 4&4 \end{bmatrix}$$

Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{\#}=M^{-1}$$

Let $$A=\begin{bmatrix}2 & -1&5 \\-3 & 4&0 \end{bmatrix} \text{ and } B=\begin{bmatrix}-3 & -4&2 \\-1 & 0&-5 \end{bmatrix}$$
Find each result.
3A+2B=?
A-3B=?
Compute the indicated matrices, if possible .
A^2B
let $$A=\begin{bmatrix}1 & 2 \\3 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 0 & -1 \\3 & -3 & 4 \end{bmatrix}$$
Let the matrix A,B, C and D be as given. Find the product of the sum of A and B and the difference between C and D.
$$A=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix},C=\begin{bmatrix}-1 & 0 \\0& 1 \end{bmatrix},D=\begin{bmatrix}-1 & 0 \\0 & -1 \end{bmatrix}$$
Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and identify its principal and secondary diagonals.
$$\begin{bmatrix}1 & 0&-i \\ 0 & -2 & 4-i \\ i&4+i&3 \end{bmatrix}$$
$$\begin{bmatrix}7 & 0&4 \\ 0 & -2 & 10 \\ 4&10&5 \end{bmatrix}$$
$$A=\begin{bmatrix}2& 1&1 \\-1 & -1&4 \end{bmatrix} B=\begin{bmatrix}0& 2 \\-4 & 1\\2 & -3 \end{bmatrix} C=\begin{bmatrix}6& -1 \\3 & 0\\-2 & 5 \end{bmatrix} D=\begin{bmatrix}2& -3&4 \\-3 & 1&-2 \end{bmatrix}$$
a)$$A-3D$$
b)$$B+\frac{1}{2}$$
c) $$C+ \frac{1}{2}B$$
(a),(b),(c) need to be solved
Given the matrix
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
and suppose that we have the following row reduction to its PREF B
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.
$$A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$