Linear algebra MOC

Diagonalization

Let be a field. A square matrix is said to be diagonalizable iff

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

Properties

  1. If is diagonalizable then is diagonalizable for .
  2. The converse holds if is algebraically closed and is invertible: If is diagonalizable for some then is diagonalizable.
Proof of 1–2

If then , proving ^P1.

#state/tidy | #SemBr