David M. Russinoff
david@russinoff.com
http://www.russinoff.com

This directory contains an ACL2 formalization of elementary elementary linear algebra, a work in progress.  The books of the top-level directory are listed below.  Once completed, the content will approximate that of a rigorous introductory course in the subject.

The ordering of topics, however, does not resemble the table of contents of any textbook on the subject, which invariably initially focuses on the algebra of matrices over a field.  The main drawback of this approach is that upon reaching the topic of eigenvalues, we find that the fundamental notion of the characteristic polynomial of a matrix over a field F is defined as the determinant of a certain matrix over the polynomial ring F[X].  We are then faced with the inconvenience that all results up to that point have been stated and proved only in the context of matrices over a field.  The usual way of dealing with this problem is simply to ignore it and hope the reader doesn't notice it.  The only exception of which I am aware is Hoffman and Kunze, "Linear Algebra", from which I learned the subject as a college sophomore and on which this formalization is largely based.  Before addressing the characteristic polynomial, the authors acknowledge the problem and advise the reader to backtrack and verify that proofs of all the properties of matrices that are relevant to this issue, though explicitly presented in the context of a field, apply to commutative rings as well.

Neither of these solutions will work here.  Thus, we begin by proving the basic properties of matrix algebra and determinants explicitly for matrices over an arbitrary commutative ring with unity.  We define such a ring by means of an encapsulated set of constrained functions: the binary ring operations, the additive and multiplicative identities as constant functions, and a predicate the recognizes ring elements.  Theorems corresponding to the ring axioms are exported from the encapsulation.  Any theorems subsequently proved about these constrained functions may be applied to any ring of interest by functional instantiation.

Thus, the first 3 books deal with matrices over a commutative ring with unity:

  * ring
    - Encapsulation of the axioms for a commutative ring with unity
    - Properties of lists of ring elemants

  * rmat
    - Matrices over a ring
    - Transpose of a matrix
    - Matrix algebra: matrix addition, scalar multiplication, and matrix multiplication

  * rdet
    - Determinant of a square matrix over a ring
    - Uniqueness of the determinant as an alternating n-linear function
    - Multiplicativity of the determinant
    - Cofactor expansion and the classical adjoint

An arbitrary field is similarly defined by a separate set of constrained functions, the only difference being an additional axiom that guarantees the existence of multiplicative inverses.  The next 3 books simply contain repetitions of the above results based on these functions, i.e., in the context of matrices over an arbitrary field.  In principle, these could all be derived from the above results by functional instantiation, but it was found to be easier just to repeat the proofs:
  
  * field

  * fmat

  * fdet

The next book, which depends on the last 3, deals with topics that apply only to matrices over a field:

  * reduction
    - Elementary row reduction
    - Row reduction as matrix multiplication
    - Invertibility
    - Systems of linear equations and Cramer's rule

In the book "rational", which is incompatible with the other books, we replace the constrained functions in field.lisp with the corresponding rational functions and in this context, we reproduce the function definitions in field.lisp, fmat.lisp, fdet.lisp, and reduction.lisp.  The result is that these definitions are now executable and may be used for the purpose of illustration.

Several additional books are planned.  The first 2 are based on the theory of matrices oever a field:

  * vector spaces

  * linear transformations

Finally, we shall address the topic of eigenvalues and the characteristic polynomial.  This will involve the theory of matrices over a field as well as that of matrices over a ring, applied to the ring of polynomials over a field:

  * ring of polynomials over a field

  * eigenvalues and the characteristic polynomial
