Undergrad Finding the matrix inverse by diagonalisation

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To find the matrix inverse by diagonalization, the relationship D = P^(-1)AP is used, where D is a diagonal matrix. It follows that D^(-1) = P^(-1)A^(-1)P, as the inverse of a diagonal matrix is straightforward to compute. The identity matrix confirms that A^(-1) can be expressed as PD^(-1)P^(-1). This method is validated by the ease of inverting diagonal matrices. The discussion emphasizes the importance of understanding these relationships in matrix algebra.
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How would you go about doing this, I see it so often quoted as a method, but no-where can I find an example

This is what I was thinking

D=P^(-1)AP

Would it then follow that D^(-1)=P^(-1)A^(-1)P ?

My reasoning being:

DD^(-1)= P^(-1)APP^(-1)A^(-1)P
identity matrix= P^(-1)AA^(-1)P=identity matrix

and hence

A^(-1)=PD^(-1)P^(-1)

Just wondering if this is what they meant or if I've completely missed the point

Manny thanks in advance :))
 
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Yes, because finding the inverse of a diagonal matrix with all-non-zero diagonal is very easy.
 
blue_leaf77 said:
Yes, because finding the inverse of a diagonal matrix is very easy.
Fab - thank you. I just wanted to check :)
 

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