Finding diagonal matrix D and invertible matrix C such that $A=C^{-1}DC$

Question

Here is the full question:

Consider the matrix: \begin{equation*} A= \begin{pmatrix} -3 & 5 \\ -2 & 4 \end{pmatrix} \end{equation*} Find a diagonal matrix $D$ and an invertible matrix $C$ such that $A =CDC^{-1}$. Deduce $A^3$. Do not find $C$. Make some conclusions.

My question is how am I supposed to find $D$ without finding C knowing that $D=C^{-1}AC$. If I could use $C$ then I would find the eigenvectors of $A$ to get C but the question says I shouldn't find it.

Answer 1

Whether or not you’re meant to diagonalize $A$ explicitly (the problem statement as you have it in your question says both “Find $C$” and “Do not find $C$;” I have no idea what to make of that), in order to find $D$ you can’t get around computing eigenvalues of $A$.

However, it’s possible to compute $A^3$ without going even that far by applying the Cayley-Hamilton theorem. Let the characteristic polynomial of $A$ be $p(x)$. Then we can divide $x^3$ by this polynomial to get $x^3=p(x)q(x)+r(x)$. Since $p(A)=0$, we then have $A^3=r(A)$, which, since $p$ is quadratic, will be of the form $aI+bA$. So, you just need to compute the characteristic polynomial and find the remainder when you divide $x^3$ by it.

Answer 2

Step 1: Find the eigenvalues and vectors. Step 2: What is D? (what is C as a consequence)? Step 3: If $A = CDC^{-1}$, what happens when you do $A^2$? $A^3$?

Answer 3

This is an eigenvalue decomposition problem (https://en.m.wikipedia.org/wiki/Eigendecomposition_of_a_matrix). $$\bf A v = \lambda v $$ where $\lambda $ is an eigenvalue, and $v$ an eigenvector. Then $$\bf A C = DC $$ Where C is a matrix made of eigenvectors, and D is a diagonal matrix whose diagonals are the eigenvalues of $\bf A$. Then $$\bf A = C D C^{-1}$$

Then it’s quite easy to find $A^3$.