Maths in Lean: linear algebra #

Semimodules, Modules and Vector Spaces #

algebra.module #

This file defines the typeclass module R M, which gives an R-module structure on the type M. An additive commutative monoid M is a module over the (semi)ring R if there is a scalar multiplication • (has_scalar.smul) that satisfies the expected distributivity axioms for + (in M and R) and * (in R). To define a module R M instance, you first need instances for semiring R and add_comm_monoid M. By splitting out these dependencies, we avoid instance loops and diamonds.

In general mathematical usage, a module over a semiring is also called a semimodule, and a module over a field is also called a vector space. We do not have separate semimodule or vector_space typeclasses because those requirements are more easily expressed by changing the typeclass instances on R (and M). In this document, we'll use "module" as the general term for "semimodule, module or vector space" and "ring" as the general term for "(commutative) semiring, ring or field".

Let m be an arbitrary type, e.g. fin n, then the typical examples are: m → ℕ is an ℕ-semimodule, m → ℤ is a ℤ-module and m → ℚ is a ℚ-vector space (outside of type theory, these are known as ℕ^m, ℤ^m and ℚ^m respectively). A ring is a module over itself, with • defined as * (this equality is stated by the simp lemma smul_eq_mul). Each additive monoid has a canonical ℕ-module structure given by n • x = x + x + ... + x (n times), and each additive group has a canonical ℤ-module structure defined similarly; these also apply for (semi)rings.

The file linear_algebra.linear_independent defines linear independence for an indexed family in a module. To express that a set s : set M is linear independent, we view it as a family indexed by itself, written as linear_independent R (coe : s → M).

The file linear_algebra.basis defines bases for modules.

The file linear_algebra.dimension defines the rank of a module as a cardinal. We also use rank for the dimension of a vector space, since the dimension is always equal to the rank. (In fact, rank is currently only defined for vector spaces, as the cardinality of a basis. A definition of rank for all modules still needs to be done.) The rank of a linear map is defined as the dimension of its image. Most definitions in this file are non-computable.

The file linear_algebra.finite_dimensional defines the finrank of a module as a natural number. By convention, the finrank is equal to 0 if the rank is infinite.

Matrices #

data.matrix.basic #

The type matrix m n α contains rectangular, m by n arrays of elements of the type α. It is an alias for the type m → n → α, under the assumptions [fintype m] [fintype n] stating that m and n have finitely many elements. A matrix type can be indexed over arbitrary fintypes. For example, the adjacency matrix of a graph could be indexed over the nodes in that graph. If you want to specify the dimensions of a matrix as natural numbers m n : ℕ, you can use fin m and fin n as index types.

A matrix is constructed by giving the map from indices to entries: (λ (i : m) (j : n), (_ : α)) : matrix m n α. For matrices indexed by natural numbers, you can also use the notation defined in data.matrix.notation: ![![a, b, c], ![b, c, d]] : matrix (fin 2) (fin 3) α. To get an entry of the matrix M : matrix m n α at row i : m and column j : n, you can apply M to the indices: M i j : α. Lemmas about the entries of a matrix typically end in _val: add_val M N i j : (M + N) i j = M i j + N i j.

Matrix multiplication and transpose have notation that is made available by the command open_locale matrix. The infix operator ⬝ stands for matrix.mul, and a postfix operator ᵀ stands for matrix.transpose.

When working with matrices, a vector means a function m → α for an arbitrary fintype m. These have a module (or vector space) structure defined in algebra.module.pi consisting of pointwise addition and multiplication. The distinction between row and column vectors is only made by the choice of function. For example, mul_vec M v multiplies a matrix with a column vector v : m → α and vec_mul v M multiplies a row vector v : m → α with a matrix. If you use mul_vec and vec_mul a lot, you might want to consider using a linear map instead (see below).

Permutation matrices are defined in data.matrix.pequiv.

The determinant of a matrix is defined in linear_algebra.determinant.

The adjugate and for nonsingular matrices, the inverse is defined in linear_algebra.matrix.nonsingular_inverse.

The type special_linear_group m R is the group of m by m matrices with determinant 1, and is defined in linear_algebra.special_linear_group.

Linear Maps and Equivalences #

algebra.module.linear_map #

The type M →[R]ₗ M₂, or linear_map R M M₂, represents R-linear maps from the R-module M to the R-module M₂. These are defined by their action on elements of M. The type M ≃[R]ₗ M₂, or linear_equiv R M M₂, is the type of invertible R-linear maps from M to M₂.

The equivalence between matrices and linear maps is formalised in linear_algebra.matrix.to_lin. to_lin shows that matrix.mul_vec is a linear equivalence between matrix m n R and (n → R) →[R]ₗ (m → R). In addition, linear_map.to_matrix takes a basis ι for M₁ and κ for M₂ and gives the equivalence between R-linear maps between M₁ and M₂ and matrix ι κ R. If you have an explicit basis for your maps, this equivalence allows you to do calculations such as getting the determinant.

The difference between matrices and linear maps is that matrices are in their essence an array of entries (which incidentally allows actions such as matrix.mul_vec), while linear maps are in their essence an action on vectors (which incidentally can be represented by a matrix if we have a finite basis). If you want to do computations, a matrix is a better choice. If you want to do proofs without computations, a linear map is a better choice.

The type general_linear_group R M is the group of invertible R-linear maps from M to itself. general_linear_equiv R M is the equivalence between general_linear_group and M ≃[R]ₗ M. special_linear_group.to_GL is the embedding from the special linear group (of matrices) to the general linear group (of linear maps).

The dual space, consisting of linear maps M →[R]ₗ R, is defined in linear_algebra.dual.

Bilinear, Sesquilinear and Quadratic Forms #

linear_algebra.bilinear_form #

For an R-module M, the type bilin_form R M is the type of maps M → M → R that are linear in both arguments. The equivalence between bilin_form R M and maps M →ₗ[R] M →ₗ[R] R that are linear in both arguments is called bilin_linear_map_equiv. A matrix M corresponds to a bilinear form that maps vectors v and w to row v ⬝ M ⬝ col w. The equivalence between bilin_form R (n → R) and matrix n n R is called bilin_form_equiv_matrix.

linear_algebra.sesquilinear_form #

For an R-module M and I : R →+* R, the type M →ₗ M →ₛₗ[I] R is the type of maps M → M → R that are linear in the first argument and that in the second argument are I-semilinear. Semilinearity of f with respect to a ring homomorphism I means the following equation hold: f x (a • y) = I a * f x y.

linear_algebra.quadratic_form #

For an R-module M, the type quadratic_form R M is the type of maps f : M → R such that f (a • x) = a * a * f x and λ x y, f (x + y) - f x - f y is a bilinear map.

Up to a factor 2, the theory of quadratic and bilinear forms is equivalent. bilin_form.to_quadratic_form f is the quadratic form given by λ x, f x x. quadratic_form.associated f is the bilinear form given by λ x y, ⅟2 * (f (x + y) - f x - f y) (if there is a multiplicative inverse of 2). quadratic_form.to_matrix and matrix.to_quadratic_form are the maps between quadratic forms and matrices.