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## Main Question or Discussion Point

I wanted to know what the usage of the following could be :

Let ##A\in M_{n\times n}(K)## a matrix over the field K.

Suppose we look for ##x,\lambda\in M_{n\times 1}(K)## such that

$$Ax=(\lambda_i x_i)$$

Hence instead of having a global eigenvalue we would have local ones.

I know the characteristic polynomial gives a relationship between the components of ##\lambda##.

What results are known about this problem, and can they have interpretations like quantum mechanics has with usual diagonalisation ?

Let ##A\in M_{n\times n}(K)## a matrix over the field K.

Suppose we look for ##x,\lambda\in M_{n\times 1}(K)## such that

$$Ax=(\lambda_i x_i)$$

Hence instead of having a global eigenvalue we would have local ones.

I know the characteristic polynomial gives a relationship between the components of ##\lambda##.

What results are known about this problem, and can they have interpretations like quantum mechanics has with usual diagonalisation ?

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