Finding the matrix inverse by diagonalisation

In summary, the conversation discusses finding the inverse of a diagonal matrix and the ease of doing so. The example provided shows how finding the inverse can be simplified using a specific method. The person asking also confirms their understanding and thanks the other person.
  • #1
Physgeek64
247
11
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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  • #2
Yes, because finding the inverse of a diagonal matrix with all-non-zero diagonal is very easy.
 
  • #3
blue_leaf77 said:
Yes, because finding the inverse of a diagonal matrix is very easy.
Fab - thank you. I just wanted to check :)
 

Related to Finding the matrix inverse by diagonalisation

1. How do you find the matrix inverse by diagonalisation?

To find the matrix inverse by diagonalisation, you first need to diagonalise the matrix by finding its eigenvalues and eigenvectors. Then, you can use the diagonalised matrix to easily find the inverse using a specific formula.

2. What is the formula for finding the inverse of a diagonalised matrix?

The formula for finding the inverse of a diagonalised matrix is: inverse = V * D-1 * V-1, where V is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues.

3. Can any matrix be diagonalised?

No, not all matrices can be diagonalised. Only square matrices with distinct eigenvalues can be diagonalised.

4. What is the advantage of finding the matrix inverse by diagonalisation?

The advantage of finding the matrix inverse by diagonalisation is that it is a much simpler and more efficient method compared to traditional methods, especially for larger matrices. It also provides a clear understanding of the matrix structure and can be used for solving systems of equations.

5. Can the matrix inverse by diagonalisation be used for non-invertible matrices?

No, the matrix inverse by diagonalisation can only be used for invertible matrices. If a matrix is non-invertible, it will not have distinct eigenvalues and therefore cannot be diagonalised.

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