To answer your second question, which is in your supplemental answer.
See my blog article for more details. https://www.dsprelated.com/showarticle/754.php
I like to use $i$, instead of $j$.
The underlying rule, which comes from the exponential nature of the unit circle, is that when you multiply two complex numbers their angles get added. Thus the angle to $i$ is $90^o$ ($\pi/2$ radians), so every time you multiply by $i$ you rotate the complex number by a quarter circle. The other rule about multiplying complex numbers is that the magnitude of the product is the product of the magnitude. Since $|i|=1$ it doesn't stretch any number you multiply it by.
You can rotate any complex number by $p$ quarter circles by multiplying it by $i^p$. You can pick any point on the unit circle, and if you multiply a complex number by the value of the complex number at that point, you will rotate the first complex value by the angle to the point.
Again, read my blog article on this. It builds up to Euler's equation using simple to understand algebra.