ABSTRACT
This chapter discusses the class of matrices that represent a particular linear operator. The geometric multiplicity of an eigenvalue is the dimension of the eigenspace of the eigenvalue. The algebraic multiplicity of an eigenvalue λ is the exponent of the factor in the characteristic polynomial. The theorems mentioned earlier say that each linear transformation on a vector space gives rise to an equivalence class of matrices, and that different linear transformations give rise to different equivalence classes of matrices. The chapter shows that similar matrices have the same characteristic polynomial, and hence the same eigenvalues and the eigenvalues have the same algebraic multiplicity.
