Does the order matter while writing a Conversion Matrix?

Question

I have two eigen vectors \begin{pmatrix} 1 \\0 \end{pmatrix} and \begin{pmatrix} 0 \\2 \end{pmatrix}. So will the eigen basis be \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} or \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix} ?

Answer 1

Suppose $A$ is the original matrix whose eigenvectors you computed. Suppose that $\xi_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $ is an eigenvector corresponding to the eigenvalue $\lambda_1$, while $\xi_2 = \begin{pmatrix} 0 \\ 2 \end{pmatrix} $ correpsonds to the eigenvalue $\lambda_2$. Define the following matrices: \begin{equation} P = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \quad \text{and} \quad Q= \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}. \end{equation}

The difference between $P$ and $Q$ is the effect they have on the diagonal representations of $A$: \begin{equation} P^{-1} A P = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix} \quad \text{whereas} \quad Q^{-1}AQ = \begin{pmatrix} \lambda_2 & 0 \\ 0 & \lambda_1 \end{pmatrix}. \end{equation}

So, the answer to your question is: it depends on what you mean by "matter". If you mean "does it make a difference", then the answer is yes, because as I mentioned above, the eigenvalues will be swapped in the diagonal representation of $A$. But, if you mean "is there any inherent reason to choose one over the other?" the answer is no; you just need to be aware of where the eigenvalues go.

---

Added Remark:

Just a comment about your terminology: you said "will the eigen basis be ...", but then you proceed to list the matrices $P$ and $Q$ I defined. I hope you know that $P$, $Q$ are called the "change of basis/coordinate matrix", whereas the eigenbasis is the basis of eigenvectors: $\{\xi_1, \xi_2\}$.

If you want to be more specific, you could speak of the "ordered eigenbasis" $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_1\}$, to emphasise that you want to keep track of the order.

I only brought this up for your knowledge, if you just happened to be a little imprecise and you know the distinction, then of course ignore this remark.

Answer 2

The order of the eigenvectors doesn’t matter per se. However, there might be other constraints in play. For instance, it’s common for the eigenvalues in the diagonal matrix to be arranged in either descending or ascending order. In that case, the eigenvectors must be arranged to match.