What is the geometrical interpretation of this operation:

Multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$


multiplication by −i = rotate by −π/2

  • $\begingroup$ Pick up a complex number and multiply it by that and see what happens ! $\endgroup$
    – alkabary
    May 22 '15 at 2:31
  • 3
    $\begingroup$ You can put that complex number in polar form. It may help with the geometric interpretation of the problem. $\endgroup$
    – MathNewbie
    May 22 '15 at 2:32
  • $\begingroup$ Or even just plot the number $z = \frac{1-i}{\sqrt2}$ on the complex plane, and remember that is the result of multiplying $1$ by $z$. More generally, see Geometric interpretation of the multiplication of complex numbers? $\endgroup$
    – David K
    May 22 '15 at 2:36

First find the magnitude $r = \left|\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right| = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1$.

So multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ does not change the magnitude.

Next find the argument by $\tan\theta = \frac{-1/\sqrt{2}}{1/\sqrt{2}} = -1 \implies \theta = -\frac{\pi}{4}$.

Thus multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ represents a rotation by $\frac{\pi}{4}$ clockwise.









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