Smith canonical form, similarity canonical form and Jorden canonical form of matrix

Three standard forms

• The equivalent standard form is a combination of invariant factors. The first r elements represent invariant factors. When the matrix is ​​not full rank, there will be zero elements on the diagonal
• The rational standard form corresponds to the invariant factor, and the coefficient features of each order of the invariant factor are extracted to form a rational block
• The Jorden standard form corresponds to the elementary factors, and the eigenvalues ​​and orders of the elementary factors are extracted to form the Jorden block

Equivalent normal form

The Smith canonical form of a matrix is ​​also called the equivalent canonical form, and the elements are invariant factors. The equivalent canonical form of a matrix
in the

form of 0 is itself.
The equivalent canonical form of a polynomial matrix is ​​unique
. When the matrix is ​​not full rank, the equivalent canonical form 0 elements will appear on the diagonal

rational normal form

Jorden canonical form