Find an invertible matrix $P$ and a diagonal matrix $D$ such that $P^{−1}AP=D$.

Question

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Answer 1

You can do this in steps:

1. Using characteristic polynomials, show that $A$ has eigenvalues $-3,-2,2$.

2. Find the eigenvectors of $A$ corresponding to the eigenvalues.

3. Set $P$ to be the matrix whose columns are eigenvectors of $A$. Set $D$ to have eigenvalues on its diagonal.

Answer 2

Eigenvalue decomposition of $A$ is $PDP^{-1}$. $D$ is a diagonal matrix with diagonal elements equal to the eigenvalues of $A$, and $P$ is the corresponding matrix whose clumns are the eigenvectors of $A$. The reason is that $$AP=PD$$ which results in $$A=PDP^{-1}$$

You can use it to get $$D=P^{-1}AP$$