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|contribs)} 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|contribs)} 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|contribs)} 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|contribs)} 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|contribs)} 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|contribs)} 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|contribs)} 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. [1], [2], [3], [4], [5]). 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|contribs)} 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|contribs)} 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|contribs)} 12:00, 16 May 2009 (UTC)

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