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From Wikipedia:

A generalized eigenvalue problem (2nd sense) is the problem of finding a vector v that obeys $$ A\mathbf{v} = \lambda B \mathbf{v} \quad \quad $$ where $A$ and $B$ are matrices.

I was wondering if $A$ and $B$ are required to be square matrices? The definition doesn't seem to require this, but the next sentence does

The possible values of $λ$ must obey the following equation $$ \det(A - \lambda B)=0.\, $$

Thanks!

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Well, also regular eigenvectors are defined for square matrices, otherwise things may hardly make sense (exercise: try with a $\,2\times 3\,$ matrix...) –  DonAntonio Nov 25 '12 at 2:43
    
Regular ones do, but the definition for generalized ones seems not. –  Tim Nov 25 '12 at 2:47
    
Well it seems yes, at least according to Wiki... –  DonAntonio Nov 25 '12 at 2:48
    
You can always complete $A$ and $B$ to square matrices by appending zero blocks to them (and in the case of "tall" matrices, also append a zero vector to $v$). So I don't see much difference between the square and non-square cases. –  user1551 Nov 25 '12 at 10:28
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1 Answer

Matrices A and B need not be square. But general algorithms does not work there. Please refer to papers where there is work done on non square matrices. Try searching generalized eigenvalue problem for non square matrices.

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