Diagonalization

Let $𝕂$ be a field. A square matrix $𝐴 \in M 𝑛, 𝑛 (𝕂)$ is said to be diagonalizable iff

𝐴 = 𝑃 𝐷 𝑃 - 1

for some diagonal matrix $𝐷$ and some invertible matrix $𝑃$ . The diagonal entries of $𝐷$ are then precisely the eigenvalues of $𝐴$ .

Properties

If $𝐴$ is diagonalizable then $𝐴 𝑛$ is diagonalizable for $𝑛 \in ℕ$ .
The converse holds if $𝕂$ is algebraically closed and $𝐴$ is invertible: If $𝐴 𝑛$ is diagonalizable for some $𝑛 \in ℕ ∖ c h a r (𝕂) ℕ$ then $𝐴$ is diagonalizable.

Proof of 1–2

If $𝐴 = 𝑃 𝐷 𝑃 - 1$ then $𝐴 𝑛 = 𝑃 𝐷 𝑛 𝑃 - 1$ , proving ^P1.

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