Talk:Jacket matrix

Example?
The current example is:

J_4 = \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\  1 & -2 & 2 & -1 \\   1 & 2 & -2 & -1 \\   1 & -1 & -1 & 1 \\  \end{array} \right], J_4^{-1} = {1 \over 4} \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\  1 & -{1 \over 2} & {1 \over 2} & -1 \\ 1 & {1 \over 2} & -{1 \over 2} & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right]. $$

However, the defining property of a jacket matrix is given by:


 * $$\ A^{\mathrm{*}}=(a_{ji}^{-1})$$

These currently don't agree (because of the overall scaling by 1/4). Should the property actually read:


 * $$\ A^{\mathrm{*}}=k(a_{ji}^{-1})$$

?? Oli Filth(talk 19:14, 6 April 2009 (UTC)


 * You missed part of the definition. It says:
 * $$ A^*A = AA^* = nI_n. \, $$
 * It does not say:
 * $$ A^*A = AA^* = I_n. \, $$
 * You need to read the whole sentence. Michael Hardy (talk) 21:15, 7 April 2009 (UTC)


 * Maybe I'm being slow, but I'm not sure how that validates the "discrepancy" I think exists above! If I understand the notation:
 * $$\ A^{\mathrm{*}}=(a_{ji}^{-1})$$
 * it's saying that each element of the conjugate transpose is the inverse of the corresponding element of the original matrix. This is clearly not the case in the example.  Oli Filth(talk 22:32, 7 April 2009 (UTC)

Definition again
Right, I see someone's fixed the problem I mentioned above. However, the "different form" also worries me:


 * $$ \sum_{i=1}^n {a_{u,i} \over a_{v,i}}=0\text{ for }u,v\in\{1,\dots,n\},\ u \neq v. $$

According to the first condition, this is a necessary condition, but not a sufficient one. Therefore, the two definitions are not equivalent. Presumably it should also include the fact that the summation should equal n when u=v? Oli Filth(talk 12:29, 9 April 2009 (UTC)


 * I see someone's now removed the alternative definition from the lead. If that's because it's wrong, it also needs removing from the "Properties" section.  Oli Filth(talk 09:06, 10 April 2009 (UTC)


 * Actually, nevermind! I've just realised that's it's self-evident what the summation is when u = v.  ($$\frac{a_{u,i}}{a_{v,i}} = 1$$).  Oli Filth(talk 09:16, 10 April 2009 (UTC)

Actually, no, hang on. That alternative definition is dangerous. Is there anything to prevent these matrices having zero-valued elements? If not, then the alternative definition will be meaningless. Oli Filth(talk 09:31, 10 April 2009 (UTC)

Fields
Does the jacket matrix work over fields in general, or only over the specific examples mentioned? If it's the former, all that need be said is:


 * "whose entries are non-zero and from a field."

If it's the latter, it should say:


 * "whose entries are from a real field, a complex field or a finite field."

Oli Filth(talk 10:30, 11 April 2009 (UTC)

Constraints for orthogonal and unitary matrices
In the articles for Orthogonal matrix and Unitary matrix, the conditions are clearly stated as $$QQ^T = I$$ and $$UU^* = I$$, respectively. In other words, the column vectors form an orthonormal basis, hence:


 * $$\sum_{i=1}^n a_{u,i}^2 = 1$$

and


 * $$\sum_{i=1}^n |a_{u,i}|^2 = 1$$

This definition tallies with several sources (e.g., , , , ). Now, perhaps their are alternative definitions of orthogonal and unitary matrices out there, but for internal consistency, the definitions on this page need to match those elsewhere in Wikipedia. Oli Filth(talk 10:55, 12 April 2009 (UTC)


 * A possible compromise is to simply remove the definitions. What purpose do they serve on a page about jacket matrices?  Oli Filth(talk 01:38, 13 April 2009 (UTC)

Vandermonde matrix?
How is the jacket matrix a Vandermonde matrix? The example currently in this article certainly isn't one! Oli Filth(talk 12:00, 16 May 2009 (UTC)

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