Simple Introduction to Eigenvalue and Similarity Form in Several Situations

The eigenvalues and similarity theory of matrices have wide applications in various branches of mathematics. For a general matrix, a simple and usable standard form and an effective method to transform it into the standard form are required. Based on certain theoretical foundations, including some narration on solution space, linear independence of eigenvectors, elementary matrices, and Kronecker product, the process involves triangulation, diagonalization, Jordan standard form, and a convenient similarity classification method over general number fields. Some introductions and derivations are provided for similar standard forms, leading to the conditions for diagonalizability of general matrices and other standard forms for some situations when diagonalization is not possible. This also gives some deeper conceptions like determinant, divisor, and zero polynomial. For some special forms of matrices, there are stronger conclusions. Additionally, applications of related methods in problems are presented, such as matrix exponentiation, and in other aspects, like solving differential equations and linear sequence problems.