def my$\_$transf$($A$)$: # A $=$ np$.$array$([[$b$,$ c$,...],[$d$,$ e$,...],[...]])$

rows, cols $=$ A$.$shape

A$\_$transpose $=$ np$.$zeros$(($cols$,$ rows$),$ dtype$=$A$.$dtype$)$

for i in range$($rows$)$:

for j in range$($cols$)$:

A$\_$transpose$[$j$,$ i$]=$ A$[$i$,$ j$]$

result $=$ A $+$ A$\_$transpose

return result

import scipy.linalg as pa

def eigprod$($A$,$ i$,$ j$)$: # A is a square matrix$($mXm$)$

eigval, eigvec $=$ pa$.$eigh$($A$)$

# Extracting the i$-$th and j$-$th eigenvectors

v$\_$i $=$ eigvec$[$:$,$ i$]$

v$\_$j $=$ eigvec$[$:$,$ j$]$

# Calculating dot product of the two eigenvectors$($v$\_$i$,$v$\_$j$)$

dot$\_$product $=$ np$.$dot$($v$\_$i$,$ v$\_$j$)$

return dot$\_$product

Construct $`$A$`$ as a $$4\backslash$times $4$$ matrix of random numbers $($using poisson distribution$).$ Check the values of $`$eigprod$($my$\_$transf$($A$),$i$,$j$)`$ for different values of $i$ and $j$$,$ and different random matrices $A$$.$ You only need to show one matrix and value of $$($i$,$j$)$$ in your answer. What do you notice?, Explain the reason for what you noticed. Manipulating the following equation:

$$v$\_2($R$)^$T $\backslash$times $($R$^$T $-$ R$)\backslash$times v$\_3($R$)$$$

$($where $$\backslash$times$ denotes matrix multiplication$)$ might help