to get values for . Recall 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.
is an eigenvalue of A is singular
trace of a Matrix is the sum of diagonal
Characteristic Polynomial
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; , which is saying how many eigenvector solutions does this eigenvalue have (recall is number of free vars in )
