I What's a non-eigenvalue called?

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Hi PF!

So if this ##L[\phi] = M[\phi]## is called an eigenvalue problem (where ##L,M## are operators, could be differential or matrices) then what is this called ##L[\phi] = M##?
 

fresh_42

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Hi PF!

So if this ##L[\phi] = M[\phi]## is called an eigenvalue problem (where ##L,M## are operators, could be differential or matrices) then what is this called ##L[\phi] = M##?
If ##(L-M).\Phi = 0## then ##\Phi## is an eigenvector of ##L-M## to the eigenvalue ##0##, i.e. an element in the kernel in case ##L,M## are linear.

Now, what is where from? Since ##L\Phi = M## makes no sense, they aren't in the same vector space anymore!

The general situation is a bit complexer than white and black: see e.g. https://www.physicsforums.com/insights/hilbert-spaces-relatives-part-ii/
 
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If ##(L-M).\Phi = 0## then ##\Phi## is an eigenvector of ##L-M## to the eigenvalue ##0##, i.e. an element in the kernel in case ##L,M## are linear.

Now, what is where from? Since ##L\Phi = M## makes no sense, they aren't in the same vector space anymore!

The general situation is a bit complexer than white and black: see e.g. https://www.physicsforums.com/insights/hilbert-spaces-relatives-part-ii/
I guess what I'm wondering is, what is this equation formally called ##A x = b##? Clearly it's not an eigenvalue problem.
 

fresh_42

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I guess what I'm wondering is, what is this equation formally called ##A x = b##? Clearly it's not an eigenvalue problem.
Again, depends on what is where from. Usually, i.e. if written this way, it is just an inhomogeneous system of linear equations. This remains the case if the vector space for the coefficients is an ordinary number field or a Hilbert space of functions and ##A## a differential operator.

As you have posed this questions under differential equations, then this imposes restrictions on ##x##. A linear differential equation of this type would be a vector ##x_i=y^{(i)}## and the coefficients of ##A## and ##b## e.g. functions ##\mathbb{R} \longrightarrow \mathbb{R}##.
 

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