Exercise 1.2.34

Question

Using $\boldsymbol{v}=\operatorname{randn}(3,1)$ in 	exttt{MATLAB}, create a random unit vector $\boldsymbol{u}=\boldsymbol{v} /\|\boldsymbol{v}\|$. Using $V=\operatorname{randn}(3,30)$ create 30 more random unit vectors $U_{j} .$ What is the average size of the dot products $\left|\boldsymbol{u} \cdot \boldsymbol{U}_{j}ight|$ ? In calculus, the average is $\int_{0}^{\pi}|\cos 	heta| d 	heta / \pi=2 / \pi$.

Prakash Anand · Accepted Answer

The commands
\begin{lstlisting}{language=matlab}
V = randn(3,30);
D = sqrt(diag(V' * V));
U = V \ D;
\end{lstlisting}
will give 30 random unit vectors in the columns of $U$. Then $\boldsymbol{u}^{\prime} * U$ is a row matrix of 30 dot products whose average absolute value should be close to $2 / \pi$.

Exercise 1.2.34

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Exercise 1.2.34

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