Interpretation of "pseudo-diagonalisation"

[jk22]

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 ?

 

[FactChecker]

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

 

[jk22]

$$\exists x\neq 0$$

 

[FactChecker]

##\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]

If the ##\lambda_i## are given, define ##\Lambda## to be the diagonal matrix with the ##\lambda_i## on the diagonal. Then your equation is equivalent to ##Ax=\Lambda x.## Another way of saying this is that ##x## is in the nullspace of ##A-\Lambda##.

Note that if all of the ##\lambda_i=\lambda## are equal then your question is the same as finding the ##\lambda##-eigenspace of ##A##. This agrees with the above because the ##\lambda##-eigenspace of a matrix ##A## is the nullspace of ##A-\lambda I##, and ##\Lambda=\lambda I## in this case.