### Abstract

We formalize an algorithm to compute the Echelon Form of a matrix. We have proved its existence over Bézout domains and made it executable over Euclidean domains, such as the integer ring and the univariate polynomials over a field. This allows us to compute determinants, inverses and characteristic polynomials of matrices. The work is based on the HOL-Multivariate Analysis library, and on both the Gauss-Jordan and Cayley-Hamilton AFP entries. As a by-product, some algebraic structures have been implemented (principal ideal domains, Bézout domains...). The algorithm has been refined to immutable arrays and code can be generated to functional languages as well.

### License

BSD License

### Topics

- Computer science/Algorithms/Mathematical
- Mathematics/Algebra

### Session Echelon_Form

- Rings2
- Cayley_Hamilton_Compatible
- Code_Cayley_Hamilton
- Echelon_Form
- Echelon_Form_Det
- Echelon_Form_Inverse
- Examples_Echelon_Form_Abstract
- Echelon_Form_IArrays
- Echelon_Form_Det_IArrays
- Code_Cayley_Hamilton_IArrays
- Echelon_Form_Inverse_IArrays
- Examples_Echelon_Form_IArrays

### Depends on

- The Cayley-Hamilton Theorem
- Gauss-Jordan Algorithm and Its Applications
- Rank-Nullity Theorem in Linear Algebra

### Used by

- Hermite Normal Form

### Auto-related entries

- QR Decomposition

## Cite

×

Echelon_Form-AFP

    @article{Echelon_Form-AFP,
      author  = {Jose Divasón and Jesús Aransay},
      title   = {Echelon Form},
      journal = {Archive of Formal Proofs},
      month   = {February},
      year    = {2015},
      note    = {\url{https://isa-afp.org/entries/Echelon_Form.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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