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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?

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  • $\begingroup$ Your N-i subscript is not compatible with i=1....N. $\endgroup$ Commented Jul 12, 2022 at 18:14
  • $\begingroup$ Hint: Learn how to identify diagonal and antidiagonal matrices. Can you represent this as a sum of one of each? $\endgroup$ Commented Jul 12, 2022 at 18:14
  • $\begingroup$ @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. $\endgroup$
    – Gze
    Commented Jul 13, 2022 at 3:46

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