# Interpretation of "pseudo-diagonalisation"

• I
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 ?

Last edited:

FactChecker
Gold Member
It seems that this would hold for some ##\lambda## for any ##x## with all non-zero elements.

• jk22 and Infrared
$$\exists x\neq 0$$

FactChecker
Gold Member
##\lambda_i = [Ax]_i/x_i## will always work as long as ##x_i \ne 0 \forall i##. Allowing different values for the ##\lambda_i## makes this property too easy to be significant.

• jk22
So what if the condition that the ##\lambda## shall be independent of the ##x## were added ?

Infrared
• 