# How can I explain the result of of multiplications by matrices

I have a vector $$x$$ with size $$N \times 1$$, it's multiplied with a $$Z$$ matrix $$N \times N$$, the resulted $$N \times 1$$ vector is $$y = Zx$$. I know that each value of $$y_i$$, where $$i = 0, 1, 2, .., N-1$$ is as follows:

$$y_i = \frac{1}{\sqrt{2}}\left [ x_i \mathrm{cos} \left ( \frac{\pi i}{N} \right ) + x_{N-i} \mathrm{sin} \left ( \frac{\pi i}{N} \right ) \right ]$$

In that case, how can I explain the matrix $$Z$$? For example can I say that $$(m,n)$$-entry of $$Z$$ be explained as follows:

$$\left [ Z \right ]_{m,n} = \frac{1}{\sqrt {2}} \sum_{j=0}^{N-1} \mathrm{cos}\left ( \frac{\pi j}{N} \right ) + \mathrm{sin}\left ( \frac{\pi j}{N} \right )$$

If not, what is the correct form of $$Z$$ which can explain that relationship?

• Your N-i subscript is not compatible with i=1....N. Jul 12, 2022 at 18:14
• Hint: Learn how to identify diagonal and antidiagonal matrices. Can you represent this as a sum of one of each? Jul 12, 2022 at 18:14
• @MarkBorgerding I corrected it, sorry $i = 0,1,...., N-1$. yes the resulted matrix can be represented as diagonal and antidiagonal matrix, but I couldn't write it a its mathematical form.
– Gze
Jul 13, 2022 at 3:46