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Old   September 24, 2013, 14:54
Default linear algebra question, spectral radius
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Shenren Xu
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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
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Old   September 24, 2013, 15:23
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Filippo Maria Denaro
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Quote:
Originally Posted by Shenren_CN View Post
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 ...
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Old   September 25, 2013, 05:18
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Shenren Xu
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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"


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Originally Posted by FMDenaro View Post
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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