Diagonalising a matrix

Question

I'm getting confused about the difference between diagonalising a matrix $A$ and finding a matrix $P$ such that $P^{-1}AP$ is diagonal.

If I want to diagonalise the matrix A, am I correct in finding the eigenvalues and then setting them onto the diagonal of a new matrix and placing zeros everywhere else?

And if I want to find a matrix $P$ such that $P^{-1}AP$ is diagonal, do I find the unit eigenvalues of the matrix $A$ and set them as the columns of the matrix $P$?

I have put unit in bold as that is an aspect of which I am particularly unsure about - whether they are to be unit or not.

Answer 1

You put the eigenvectors of $A$ as columns of $P$. If you don't use unit vectors, it'll still work out, because the $P^{0-1}$ will take care of that.

To diagonalize $A$, you find those eigenVECTORS (not values!!), put them in $P$ as columns, and compute $P^{-1}AP$, which will be diagonal, with the eigenVALUES on the diagonal; the $i$th diagonal entry will be the eigenvalue for the $i$th eigenvector (i.e., the $i$th column of $P$).