Can we find Eigenvalues for simultaneous equation?

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Eigenvalues can be found for simultaneous equations, but limitations exist primarily in the context of dimensionality. The fundamental theorem of algebra guarantees at least one complex eigenvalue for any characteristic polynomial. In finite-dimensional real vector spaces, there are invariant subspaces of degree one or two. However, these principles only apply in finite dimensions, indicating that infinite-dimensional spaces may present additional challenges. Understanding these constraints is crucial for accurately determining eigenvalues in various mathematical contexts.
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hi, please tell me what are the limitations for finding eigenvalues ?
thanks
 
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Could you be more specific? By the fundamental theorem of algebra, there always exists at least one (complex) solution to the characteristic polynomial so there will always be at least one (complex) eigenvalue, where complex means a number of the form ##a + bi##.
 
WannabeNewton said:
Could you be more specific? By the fundamental theorem of algebra, there always exists at least one (complex) solution to the characteristic polynomial so there will always be at least one (complex) eigenvalue, where complex means a number of the form ##a + bi##.
Indubitably.

And every finite dimensional real vector space has an invariant subspace of degree one OR two.
 
WannabeNewton said:
Could you be more specific? By the fundamental theorem of algebra, there always exists at least one (complex) solution to the characteristic polynomial so there will always be at least one (complex) eigenvalue, where complex means a number of the form ##a + bi##.

This is of course only true in the finite-dimensional case.
 

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