6.2 Diagonalization

What is Diagonalization

• Recall from section 4.6 the formula:

  • $[L{]}_B=P_{S\to B}[L]P_{B\to S}=P^{-1}[L]P$
  • this is one of the motivations form “similar matrices”.

• Defininition

  • Let A and B be nxn matrices. A is similar to B if there exsts a matrix $P$ such that $P^{-1}AP=B$

    • basicly just changing the matrice to a diffrent basis

Properties of Diagonalization

• Similar matrices have the same

  • Determinant

    • so we can just diagonalize to get the determinant easily

  • Eigenvalues

  • Rank

• An nxn matrix A is Diagonalizable if A is similar to a diagonal matrix that is:

  • $P^{-1}AP=D$

When is a matrix diagonalizable

• An nxn matrix A is diagonalizable if and only if there is a basis consisting of n linearly independent eigenvectors of A.

  • it is only diagonizable if theer is the same amount of eigen values as collumns/rows