Problem $4($programming$)$

This example shows how SVD can be used to find dominant modes. Given $N$ vectors, $a_1,$dots,$a_N,$

we form a matrix $A=[a_1,$dots,$a_N],$ which is $MN$ matrix. It is known that the dominant modes

can be obtained via SVD $($eigenvectors corresponding to largest singular values of matrix $A,$

$\{$:${}_i^2),$ denote them by $v_1,$dots,$v_l.$ We will show what that means for the most dominant $v_1.$ It can

be shown that

$mi{n}_v{\displaystyle \underset{j=1}{\overset{N}{}}}||a_j-($:$a_j,v$:$)v|{|}^2$

is minimal for $v=v_1,$ assuming $||v||=1.$

$($a$)$ Take $20$ random vectors with dimension $80($you can simply use $A=$rand$(80,20))$ and do

SVD and identify the most dominant eigenvectors.

$($b$)$ Find the error for the projection defined as

Err$={\displaystyle \underset{j=1}{\overset{N}{}}}||a_j-($:$a_j,v_1$:$)v_1|{|}^2.$

Compute relative error

$E=\frac{\text{E}\text{r}\text{r}}{{\displaystyle \underset{j=1}{\overset{N}{}}}||a_j|{|}^2}.$

$($c$)$ Take an arbitrary random vector $v,$ normalize it$,$ and find the corresponding projection

${\displaystyle \underset{j=1}{\overset{N}{}}}||a_j-($:$a_j,v$:$)v|{|}^2$