Linear algebra MOC

Diagonalization

Let 𝕂 be a field. A square matrix 𝐴 ∈M𝑛,𝑛⁡(𝕂) is said to be diagonalizable iff

𝐴=𝑃𝐷𝑃−1

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

Properties

1. If 𝐴 is diagonalizable then 𝐴𝑛 is diagonalizable for 𝑛 ∈ℕ.
2. The converse holds if 𝕂 is algebraically closed and 𝐴 is invertible: If 𝐴𝑛 is diagonalizable for some 𝑛 ∈ℕ ∖char⁡(𝕂)ℕ then 𝐴 is diagonalizable.

Proof of 1–2

If 𝐴 =𝑃𝐷𝑃−1 then 𝐴𝑛 =𝑃𝐷𝑛𝑃−1, proving ^P1.

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