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 |)

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
asked 2021-02-26
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 \(\| x \|\))

Answers (1)

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,
\(\|Ux\|=\sqrt{5}\)
0

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