Linear transformation is diagonalisable
If there is a basis B for V s.t. MatB,B(T) is a diagonal matrix
Square matrix is diagonalisable
There is an invertible matrix P s.t. P-1AP is diagonal
Algebraic multiplicity
Largest power k s.t. (x-lambda)^k is a factor of the character polynomial k
Geometric multiplicity
Dimension g_lambda of the eigenspace
Monic polynomial
Polynomial where the leading coefficient is 1
Minimum polynomial
Monic polynomial Mt(x) with coefficients in F of smallest degree s.t. Mt(T) = 0
Theorem 9.1.5 Equivalence
(i) T is diagonalisable
(ii) There is a basis for V consisting of eigenvectors for T
(iii) The characteristic polynomial Ct(x) is a product of linear factors and a_lambda = g_lambda for all eigenvalues
(iv) The minimum polynomial Mt(x) is a product of distinct linear factors
Easy way to show that a matrix is diagonalisable/not diagonalisable
- Find the eigenvalues of T
- Show the algebraic and geometric multiplicities are not the same
Relationship between linear factors and diagonalisability
Linear factors -> does not mean diagonalisable
Non linear factors -> non diagonalisable
Diagonalisable -> linear factors