Diagonalizable

Definition

A Linear Map $T:V\to V$ for a finite dimensional vector space is diagonalizable if its basis for $V$ consists only of eigenvectors of T.

Diagonalization Definition

To diagonalize a matrix $A$, it must be expressed as a product of matrixes: $A=PD{P}^{-1}$

• For some Invertable Matrix $P$ (The matrix of eigenvectors Column Vectors)
  • This matrix is comprised of the Eigenvectors of $A$
• For a Diagonal Matrix $D$ (The diagonal matrix of eigenvalues )
  • This matrix is comprised of the Eigenvalues of $A$ Then, it follows that: $P^{-1}AP=D$

Concepts

• Diagonal Matrices
• Hermitian
• Normal
• Unitary

Theorems

• Characterization of Diagonizability
• Diagonalize a Matrix Example
• Nice Case of Diagonalization
• Sum Characterization of Diagonalizable