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I have a set of (about 100) general real square matrices. Is it possible to determine whether none of their linear combinations has complex eigenvalues?

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Nothing special about these "general real square matrices"? –  J. M. Aug 4 '12 at 7:38
    
perhaps, they have real eigenvalues –  Boris Aug 4 '12 at 7:49
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First of all, choose a basis of the space of linear combinations, so you'll have to work with fewer matrices. –  Dario Aug 4 '12 at 10:33
    
Do the matrices have any other properties? they commute? –  i. m. soloveichik Aug 4 '12 at 13:33
    
they are no other special properties, they do not commute, some of them are singular (1 or 2 eigenvalues are zero) –  Boris Aug 4 '12 at 16:02

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