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Canonical forms of matrices

Created over 8 years ago, updated 10 days ago

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.