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December 15th, 2016, 01:19 AM   #1
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Matrix and vectors

1. In given A in size of mXn while m>n and: (A^T)*A=I.
What can i say about the eigenvalues of A*(A^T)? and why?
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December 15th, 2016, 07:24 AM   #2
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Try writing out an explicit example.
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December 20th, 2016, 01:42 PM   #3
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If your matrix is m x n dimensional, unless m=n I dont see how do you want to get the eigenvalues.... They are only possible to calculate in square matrices
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December 20th, 2016, 02:41 PM   #4
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Originally Posted by nietzsche View Post
If your matrix is m x n dimensional, unless m=n I dont see how do you want to get the eigenvalues.... They are only possible to calculate in square matrices
they want the eigenvalues of $A A^T$ which will be $m\times m$
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December 20th, 2016, 02:53 PM   #5
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they want the eigenvalues of $A A^T$ which will be $m\times m$
you're given $A^T A=I$

you want the eigenvalues of $A A^T$

$A A^T x = \lambda x$

$A^T A A^T x = A^T \lambda x = \lambda A^T x$

$I A^T x = A^T x = \lambda A^T x$

what does this say about $\lambda$ and/or $A^T x$ ?
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December 20th, 2016, 03:14 PM   #6
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I would just form $A^TA=I \implies AA^TA=AI=A$ so the columns of $A$ are eigenvectors of $(AA^T)$ with eigenvalues of...
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December 20th, 2016, 03:47 PM   #7
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Originally Posted by v8archie View Post
I would just form $A^TA=I \implies AA^TA=AI=A$ so the columns of $A$ are eigenvectors of $(AA^T)$ with eigenvalues of...
so clever
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