Canonical forms of matrices are standard representatives for equivalence or similarity classes of matrices. A canonical form provides a unique, simplified representation that reveals key structural properties.
**Canonical forms of matrices** are standard representatives for equivalence or similarity classes of matrices. A canonical form provides a unique, simplified representation that reveals key structural properties.
Important canonical forms include:
- **Rank canonical form**: $\begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ for matrix equivalence
- **Row echelon form / RREF**: for row equivalence
- **Jordan canonical form**: for similarity (every square matrix over $\mathbb{C}$ is similar to a Jordan matrix)
- **Rational canonical form**: for similarity (works over any field)
- **Hessenberg form**: nearly upper triangular, used as intermediate step
- **Diagonal form**: when a matrix is diagonalizable
- **Schur form**: upper triangular via unitary similarity
- **SVD**: $U\Sigma V^*$ via two unitary transformations
Each canonical form is adapted to a particular equivalence relation and reveals different aspects of the matrix structure.