Reducing a matrix before diagonalization

Question

Say I have a matrix $M= \begin{bmatrix} 0&8\\16&8\end{bmatrix}$ and I wish to diagonalize it. Can I reduce it to $M= \begin{bmatrix} 0&1\\2&1\end{bmatrix}$ first and then diagonalize?

Answer 1

You can do the following:

Introduce $\bar M = \frac{1}{8}M$.

If the diagonalization of $M$ is $P D P^{-1}$, then $\bar M = \frac{1}{8}PDP^{-1} = P\bar DP^{-1}$. $P\bar DP^{-1}$ is the diagonalization of $\bar M$, with $\bar D = \frac{1}{8}D$.

I.e. if you have the diagonalizaton of the "reduced" matrix, the diagonalization of the original have the same $P$ matrix and $D=8\bar D$ for the diagonal.

Answer 2

You should specify how how you wish to reduce the matrix and how you wish to diagonalize it. Suppose $P$ is an invertible matrix and $P^{-1}AP=D$, where $D$ is a diagonal matrix. Then, for a scalar $c$ $$P^{-1}(cA)P=cD \text { and $cD $ is a diagonal matrix.}$$ So, if your method of reduction is "multiplication by a non-zero scalar" and if your diagonalization is "diagonalization by similarity," the answer is "yes but the diagonal matrix is also multiplied by the same scalar."Similarly,if $P$ is an invertible matrix and $P^TAP=D$, where $D$ is a diagonal matrix it follows that, for a scalar $c$, $$P^T(cA)P=cD \text { and $cD $ is a diagonal matrix.}$$ So, if your method of reduction is "multiplication by a non-zero scalar" and if your diagonalization is "diagonalization by congruence," the answer is again "yes but the diagonal matrix is also multiplied by the same scalar."But there are many methods of reduction and many kinds of diagonalization and the relation between them varies according to which particular method of reduction and which particular definition of diagonalization you are considering.