Characteristic Equation

$det(A-\lambda I)=0$ to get values for $\lambda$. Recall $det(T)=0$ means noninvertible. If a matrix isn’t invertible, then we won’t get trivial solutions when solving. Also the idea of reducing the dimension through the transformation is relevant; squishing the basis vectors all onto the same span where the area/volume is 0. Recall eigenvectors remain on the same span despite a linear transformation.

$\lambda$ is an eigenvalue of A $⟺$ $(A-\lambda I)$ is singular

trace of a Matrix $tr(M)$ is the sum of diagonal

Characteristic Polynomial

$det(A-\lambda I)$ n degree polynomial → n roots → maximum n eigenvalues (could be repeated)

Algebraic Multiplicity

Algebraic multiplicity of an eigenvalue is how many times an eigenvalue repeatedly occurs as the root of the characteristic polynomial.

Geometric Multiplicity

• Geometric multiplicity of an eigenvalue is the number of eigenvectors associated with an eigenvalue; $dim(Nul(A-\lambda I))$, which is saying how many eigenvector solutions does this eigenvalue have (recall $dim(B)$ is number of free vars in $B$)