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This question is related to, and a follow-up for, this question. The notation here follows that of the text quoted there.

Let $\matrix M_n$ be a Vandermonde matrix of size $n$ by $n$. The columns of $\matrix M$ are orthogonal to each other, therefore each column is mapped onto a coordinate axis. $w$ is a $n^{th}$ root of unity, and $n$ is a power of $2$. The inversion formula is supposedly:

$\matrix M_n(w)^{-1} =\frac{1}{n}\matrix M_n\left(w^{-1}\right)$

The book explains why multiplying one column by another results in 1. Therefore I understand why multiplying $M(w)$ by its complex conjugate results in $1$. What is $\frac{1}{n}$ though? Why is it part of the equation and what is it?

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    Could you explain why you accepted an answer to [this question](http://math.stackexchange.com/q/148456/742) which states this result, only to now turn around and ask for an explanation of why the result is true? What you quoted in that question seems like an explanation for this! This suggests you did not understand the answer there, or that they didn't really give you the necessary tools to understand it. Why put forth the same question again, but now without all the context?2012-05-23
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    @ArturoMagidin The question you mentioned does not ask about the formula. It inquires about a related subject, but the purpose of that question is entirely different. While the previous formula inquired about the relation between vectors and a coordinate system, this question asks about a specific inversion formula. Just because both questions reference the same piece of information does not necessary mean both are inquiring about the identical piece of information.2012-05-23
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    The previous question ask for an explanation of the terminology involved, presumably so that you could understand the text in question. The text in question is, in point of fact, an answer to **this** question you are now asking: it explains *why* you get this inversion formula. If you are not able to understand that text to the point of answering this question yourself, it means that the answers there did not, in fact, provide you with the information that you asked about in a way that is useful to you.2012-05-23
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    The previous question asked about terminology used in the question, nots its implications. The text does answer the question, but I wasn't able to understand it, resulting in me asking the question. There is a profound difference between asking about the terms in a mathematical text and its implications. These questions are different, much like the difference between asking *what* a polynomial is, and *how* to solve one. Understanding the *definition* of a concept doesn't necessary mean you understand its *implication*.2012-05-23
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    You might also note that the set-up in that question assumed that $n$ was a power of $2$ and that $w$ was a complex $n$th root of unity, conditions you omitted here. As stated, the formula does not make sense, since a Vandermonde matrix is not determined by a single parameter; so absent *that* text, this question does not even make sense. Even if we were to successfully guess that $M_n(w)$ is meant to be the Vandermonde matrix associated to $1$, $w$, $w^2,\ldots,w^{n-1}$, as stated the claim is false (cont)2012-05-23
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    (cont) since, for example, the inverse of $M_3(3)$ is not $\frac{1}{3}M_3(1/3)$.2012-05-23
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    That's my point. The text **does** answer the question you are asking. The text *explains* why the inversion formula is true; the fact that you did not understand the text is the reason why you are asking this **follow-up** question. At the very least, you should link this question to the last and explain that you were unable to understand the text leading up to the formula. (Though, in my opinion, you should have continued to try to get answers to the other question that led you to actually *understand* the text.)2012-05-23
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    You are providing entirely two different criticisms. You initially criticized the question for being a duplicate and a rewording of my previous question. You are kow criticizing my question on the basis that it is mathematically flawed. While I agree with your second criticism (and therefore corrected the mistake), I disagree with the fact that it is the same as my previous question. While criticisms that correct any mistakes in the question, are appreciated, invalid points such as it being a duplicate are a waste of your time and mine2012-05-23
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    It is a waste of everyone's time perpetrated by **you** to ask a **connected** question without even referencing the previous question it connects to (and in the process making it unintelligible and incorrect). It is a waste of your responders' time, **perpetrated by you** to accept answers that do not, in fact, answer your questions. Do you really want to talk about wasting people's time? It is not an "invalid point"; it's just one you refuse to acknowledge.2012-05-23
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    My previous question asked specifically about a certain aspect about the text, which *was* answered. Considering I am now asking about a different aspect about the text, it makes sense to ask a new question. However, I have on your suggestion added a link to the previous question.2012-05-23
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    The notation is still insufficient and incorrect. An $n\times n$ Vandermonde matrix is determined by $n$ parameters, not a single one. Your notation does not explain how the single parameter works. But, since I'm just wasting your time, I'll let you muddle through it yourself.2012-05-23
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    @ArturoMagidin You seem to be taking this personally. Apologies if my comments seem harsh or insulting. I just feel this is a valid question. I will however keep your suggestion to link to relevant questions and necessary text in mind, and ensure my questions provide all the necessary information required for a solution.2012-05-23

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Note: $w$ must be a primitive $n$-th root of unity, not just any $n$-th root (for example, $M_n(1)$ does not satisfy orthogonality of the columns).

Let $A=M_n(w)$, $B=M_n(w^{-1})$, $C=AB$.

Recall the definition of matrix multiplication: $$c_{ij}=\sum_{k=1}^n a_{ik} b_{kj}$$ On the diagonal $i=j$, $a_{ik}b_{ki}=w^{ik}w^{-ik}=1$ (is this what you meant when saying that multiplication results in 1?). But since we are summing over $n$ such terms, we have $c_{ii}=n$. Hence we need the $1/n$ factor to obtain an identity matrix: $AB = n I_n$ and $A^{-1}=\frac 1 n B$.

In other words, the columns of $M_n$ are orthogonal, but not orthonormal: the squared norm of each column is $n$. You could define $M'_n(w)=\frac{1}{\sqrt n} M_n(w)$ to make the columns orthonormal and avoid this factor.