# linear algebra question, spectral radius

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 September 24, 2013, 14:54 linear algebra question, spectral radius #1 Member   Shenren Xu Join Date: Jan 2011 Location: London, U.K. Posts: 63 Rep Power: 7 Dear colleagues, Could anyone please help me on this seemingly (maybe it in fact is) simple linear algebra problem? Suppose you have non-singular matrices A and B, is the following true? \rho (A * B) =rho (A^T * B^T) \rho here means the spectral radius of the matrix. If so, how do I prove it mathematically? Regards, Shenren

September 24, 2013, 15:23
#2
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 2,614
Rep Power: 33
Quote:
 Originally Posted by Shenren_CN Dear colleagues, Could anyone please help me on this seemingly (maybe it in fact is) simple linear algebra problem? Suppose you have non-singular matrices A and B, is the following true? \rho (A * B) =rho (A^T * B^T) \rho here means the spectral radius of the matrix. If so, how do I prove it mathematically? Regards, Shenren

I am not sure your relation is true ... I just remember that rho(A*B) <= ||A|| ||B|| where ||*|| is a matrix norm ...

September 25, 2013, 05:18
#3
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Shenren Xu
Join Date: Jan 2011
Location: London, U.K.
Posts: 63
Rep Power: 7
I found the proof afterwards however, from a paper by
Prof. Mike Giles, "on the iterative solution of a discrete adjoint equation",
which can be found by googling "mike giles oxford adjoint"

Quote:
 Originally Posted by FMDenaro I am not sure your relation is true ... I just remember that rho(A*B) <= ||A|| ||B|| where ||*|| is a matrix norm ...

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