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 Post subject: about 2-norm matrix proofPosted: Sat, 26 Feb 2011 12:30:19 UTC
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Joined: Sun, 24 Jan 2010 07:19:31 UTC
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Hi, everyone!
I am puzzled by the following problem.

Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

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 Post subject: Re: about 2-norm matrix proofPosted: Sat, 26 Feb 2011 20:48:14 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Location: Austin, TX
jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

What is that dot? It cannot be the dot product as one of those is a vector and the other is a covector.

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 Post subject: Re: about 2-norm matrix proofPosted: Sat, 26 Feb 2011 23:12:32 UTC
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Joined: Wed, 1 Oct 2003 04:45:43 UTC
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jane08 wrote:
Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

where is the angle between v and v^T.

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 Post subject: Re: about 2-norm matrix proofPosted: Sat, 26 Feb 2011 23:48:25 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Matt wrote:
jane08 wrote:
Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

where is the angle between v and v^T.

Matt, that doesn't make sense, isn't the same kind of object

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 Post subject: Posted: Sun, 27 Feb 2011 00:19:29 UTC
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I assume that jane08 means the dot product of v with itself.

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 Post subject: Re: about 2-norm matrix proofPosted: Sun, 27 Feb 2011 03:50:40 UTC
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Joined: Sun, 24 Jan 2010 07:19:31 UTC
Posts: 19
jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

What is that dot? It cannot be the dot product as one of those is a vector and the other is a covector.

means that the column vector times the row vector , and this result is a rank-1 matrix.

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 Post subject: Re: about 2-norm matrix proofPosted: Sun, 27 Feb 2011 03:53:16 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
jane08 wrote:
jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that is an orthonormal vector, and

Can we conclude that ? If so, could you give me some hints?

What is that dot? It cannot be the dot product as one of those is a vector and the other is a covector.

means that the column vector times the row vector , and this result is a rank-1 matrix.

What product are you using?

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 Post subject: Posted: Sun, 27 Feb 2011 04:48:13 UTC
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It looks like I was mistaken.
It's the matrix product of an matrix and a matrix, in which case:

Last edited by Matt on Sun, 27 Feb 2011 05:02:27 UTC, edited 1 time in total.

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 Post subject: Posted: Sun, 27 Feb 2011 05:01:05 UTC
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Matt wrote:
It looks like I was mistaken.
It's the matrix product of an matrix and a matrix, in which case:

Are we sure this isn't the operator 2-norm?

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 Post subject: Posted: Sun, 27 Feb 2011 05:09:46 UTC
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If this is the operator norm defined by:

then the answer is still 1, as the product is maximum when x = v.

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 Post subject: Posted: Sun, 27 Feb 2011 05:11:59 UTC
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Matt wrote:
If this is the operator norm defined by:

then the answer is still 1, as the product is maximum when x = v.

I agree, but I'm trying to get the original poster to be more specific, as if I hadn't asked and that WAS what he was looking for then he'd be up the creek without a paddle.

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 Post subject: Posted: Sun, 27 Feb 2011 05:20:55 UTC
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It's the OP's responsibility to point out any faulty assumptions.
If she meant the operator norm then she's free to post a correction.

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