Maths in lean : Linear Algebra #

Semimodules, Modules and Vector Spaces #

algebra.module #

This file defines the typeclass semimodule R M, which gives an R-semimodule structure on the type M, and similarly module R M and vector_space R M. An additive commutative monoid M is a semimodule over the semiring 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 semimodule 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.

A module (typeclass module) is a semimodule that additionally requires that R is a ring and M is a group. A vector space (typeclass vector_space) is a module that additionally requires that R is a field. All vector spaces are also modules, and all modules are also semimodules.

Let m be an arbitrary type, e.g. fin n, then the typical examples are: m → ℕ is an -semimodule, m → ℤ is a -module and m → ℚ is a -vector space (outside of type theory, these are known as ℕ^m, ℤ^m and ℚ^m respectively). These instances are defined in algebra.pi_instances. A (semi)ring is a (semi)module over itself, with defined as * (this equality is stated by the simp lemma smul_eq_mul).

The file linear_algebra.basis defines linear independence and bases for modules.

The file linear_algebra.dimension defines the dimension of a vector space as the minimum cardinality of a basis. The rank of a linear map is defined as the dimension of its image. Most definitions in this file are non-computable.

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 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.pi_instances 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.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 #

linear_algebra.basic #

The type M →[R]ₗ M₂, or linear_map R M M₂, represents R-linear maps from the R-module M to the R-mdule 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. linear_equiv_matrix' shows that matrix.mul_vec is a linear equivalence between matrix m n R and (n → R) →[R]ₗ (m → R). In addition, linear_equiv_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 : ring_anti_equiv R R, the type sesq_form R M I is the type of maps M → M → R that are linear in the first argument and that in the second argument are antilinear with respect to an R-antiautomorphism I. Antilinearity of f with respect to a ring antiautomorphism I means the following equations hold: f x (a • y) = I a * f x y, I 1 = 1, I (x + y) = I x + I y and I (x * y) = I y * I x.

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.