In preparation for an upcoming exam in linear algebra, I am doing some practice problems, one of which is (generally) of this form:
- Given an $n \times n$, diagonalizable matrix $A$, find invertible $Q$ and diagonal $D$ such that $Q^{-1}AQ = D$.
The following are the steps I take to do this.
- Compute the characteristic polynomial of $A$ and find its roots $\lambda_1 \dots \lambda_k$.
- For each $\lambda_i$, find linearly independent eigenvectors corresponding to $\lambda_i$; there will be as many as the multiplicity of $(x-\lambda_i)$ in the characteristic polynomial.
- Construct a $D$ whose diagonal entries are the eigenvalues; repeats appear as many times as their multiplicity.
- Construct a $Q$ whose column vectors, from left to right, are the eigenvectors corresponding to the eigenvalues as they appear along the diagonal of $D$; for repeated eigenvalues, each of the lin. independent eigenvectors should appear once in $Q$.
- Compute the inverse of $Q$. Done!
I am posting here to ask if this process can be simplified, or if this is the go-to, trusty approach I should be taking.