MHB Finding Matrix D Without Calculating P Inverse: Help Appreciated!

[tomc612]

Hi,
got a question I'm stuck on..

Write down a matrix P which will diagonalise A and write down the corresponding
diagonal matrix D, where D = P^-􀀀1AP. You do not have to calculate P^-1

Ive got all the eigenvalues and eigenvectors for A, and thus have the Matrix P, which has a determinant of -12 and thus P^-1 exists.

Question is how to do you determine D without calculating the inverse of P?

Any help appreciated

Tom

 

[Opalg]

Hi,
got a question I'm stuck on..

Write down a matrix P which will diagonalise A and write down the corresponding
diagonal matrix D, where D = P^-􀀀1AP. You do not have to calculate P^-1

Ive got all the eigenvalues and eigenvectors for A, and thus have the Matrix P, which has a determinant of -12 and thus P^-1 exists.

Question is how to do you determine D without calculating the inverse of P?

Any help appreciated

Tom

The entries on the main diagonal of $D$ are the eigenvalues of $A$. All the other entries in $D$ are zeros.

When you wrote down the matrix $P$, its columns were the eigenvectors of $A$ (in some order). When you write the diagonal elements of $D$, you should use the corresponding eigenvalues in the same order.

 

[HallsofIvy]

I am surprised that tomc612 would be given a problem like this if he had not already learned everything Opalg said!