Question

Suppose A is the matrix for T: R3 → R3 relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'.

Answer 1

Answer:

$A'=PAP^{-1}$

Step-by-step explanation:

The first that you must ask is whether or not the matrix $A$ is a diagonalizable matrix. If so, you have to find a basis of $\mathbf{R}^{3}$ consisting of eigenvectors of the matrix $A$. To do that first find the eigenvalues of the matrix and the the corresponding eigenspaces to that eigenvalues. This will give you the basis of $\mathbb{R}^{3}$ consisting of eigenvectors of A. Next apply the change of coordinates using the matrix of change of coordinates $P$ whose columns are the vectors on the new basis. So we have that

$D=A'=PAP^{-1}$

is the matrix of T respect to new basis consisting of eigenvectors of T, and it is a diagonal matrix, and its elements on the diagonal are the eigenvalues.