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PostPosted: Sat, 26 Feb 2011 12:30:19 UTC 
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Hi, everyone!
I am puzzled by the following problem.

Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

Thank you in advance!


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PostPosted: Sat, 26 Feb 2011 20:48:14 UTC 
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jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

Thank you in advance!


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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PostPosted: Sat, 26 Feb 2011 23:12:32 UTC 
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jane08 wrote:
Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

v\cdot v^T=\lVert v\rVert_2\lVert v^T\rVert_2\cos\theta

where \theta is the angle between v and v^T.


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PostPosted: Sat, 26 Feb 2011 23:48:25 UTC 
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Matt wrote:
jane08 wrote:
Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

v\cdot v^T=\lVert v\rVert_2\lVert v^T\rVert_2\cos\theta

where \theta is the angle between v and v^T.


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

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PostPosted: 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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PostPosted: Sun, 27 Feb 2011 03:50:40 UTC 
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Shadow wrote:
jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

Thank you in advance!


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



v \cdot v^T means that the column vector v times the row vector v^T , and this result is a rank-1 matrix.


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PostPosted: Sun, 27 Feb 2011 03:53:16 UTC 
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jane08 wrote:
Shadow wrote:
jane08 wrote:
Hi, everyone!
I am puzzled by the following problem.

Assume that v is an orthonormal vector, and \|v\|_2=1

Can we conclude that \|v \cdot v^T\|_2=1 ? If so, could you give me some hints?

Thank you in advance!


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



v \cdot v^T means that the column vector v times the row vector v^T , and this result is a rank-1 matrix.


What product are you using?

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PostPosted: 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 n\times 1 matrix and a 1\times n matrix, in which case:

\begin{aligned}
\lVert v\cdot v^T\rVert_2
=\sqrt{\sum_{i=1}^n\sum_{j=1}^n\lvert v_iv_j\rvert^2}
=\sqrt{\sum_{i=1}^n\lvert v_i\rvert^2\sum_{j=1}^n\lvert v_j\rvert^2}
=\sqrt{1\cdot1}
=1
\end{aligned}


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

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PostPosted: 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 n\times 1 matrix and a 1\times n matrix, in which case:

\begin{aligned}
\lVert v\cdot v^T\rVert_2
=\sqrt{\sum_{i=1}^n\sum_{j=1}^n\bigl(\lvert v_i\rvert\lvert v_j\rvert\bigr)^2}
=\sqrt{\sum_{i=1}^n\lvert v_i\rvert^2\sum_{j=1}^n\lvert v_j\rvert^2}
=\sqrt{1\cdot1}
=1
\end{aligned}


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

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

$\rule{0.1pt}{0.1pt}\quad
\lVert v\cdot v^T\rVert_2
=\max_{\lVert x\rVert_2=1}\lVert (v\cdot v^T)x\rVert_2}

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


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

$\rule{0.1pt}{0.1pt}\quad
\lVert v\cdot v^T\rVert_2
=\max_{\lVert x\rVert_2=1}\lVert (v\cdot v^T)x\rVert_2}

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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PostPosted: 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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