Linear algebra

Contents

• Creation of vector spaces
• Creation of vectors
  • From coefficients
  • Special forms
  • Coercion
• Basic properties of vector spaces
• Vector components
• Creation of matrix spaces
• Creation of matrices
  • From coefficients
  • Special forms
  • Coercion
• Basic properties of matrix spaces
• Matrix components
• Arithmetic
  • Addition
  • Scalar multiplication
  • Matrix multiplication
• Transpose, determinant, inverse

Creation of vector spaces

VectorSpace (K :: FldPadExact, n :: RngIntElt)

KSpace (K :: FldPadExact, n :: RngIntElt)

-> ModTupFld_FldPadExact

The full vector space over K of dimension n.

Creation of vectors

From coefficients

Vector (F :: FldPadExact, n :: RngIntElt, cs)

-> ModTupFldElt_FldPadExact

The vector of length n over F defined by cs.

Vector (cs :: [FldPadExactElt])

-> ModTupFldElt_FldPadExact

The vector whose coefficients are given by cs.

Special forms

Zero (V :: ModTupFld_FldPadExact)

ZeroVector (V :: ModTupFld_FldPadExact)

ZeroVector (F :: FldPadExact, n :: RngIntElt)

-> ModTupFldElt_FldPadExact

The zero vector.

Coercion

The following can be coerced to a vector in V:

• A vector in V
• A vector whose components are coercible to the base field of V
• A sequence of vector entries, all coercible to the base field

IsCoercible (V :: ModTupFld_FldPadExact, X)

-> BoolElt, Any

True if X is coercible to an element of V. If so, also returns the coerced element.

Basic properties of vector spaces

BaseField (V :: ModTupFld_FldPadExact)

-> FldPadExact

The base field of V.

BaseField (v :: ModTupFldElt_FldPadExact)

-> FldPadExact

The base field of v.

Degree (V :: ModTupFld_FldPadExact)

-> RngIntElt

If V is a subspace of K^n, returns n. That is, the number of columns in vectors in V.

Generic (V :: ModTupFld_FldPadExact)

-> ModTupFld_FldPadExact

The generic vector space containing V.

IsGeneric (V :: ModTupFld_FldPadExact)

-> BoolElt

True if V is generic, i.e. it was created as the full-dimensional vector space with default generators.

Dimension (V :: ModTupFld_FldPadExact)

-> RngIntElt

The dimension of V.

Generators (V :: ModTupFld_FldPadExact)

-> []

The generators of V.

Vector components

Nrows (v :: ModTupFldElt_FldPadExact)

Ncols (v :: ModTupFldElt_FldPadExact)

-> RngIntElt

Number of rows (always 1) and columns.

Eltseq (v :: ModTupFldElt_FldPadExact)

-> []

The components of v as a sequence.

Component (v :: ModTupFldElt_FldPadExact, i :: RngIntElt)

'@' (i :: RngIntElt, v :: ModTupFldElt_FldPadExact)

-> FldPadExactElt

The ith component of v.

Creation of matrix spaces

KMatrixSpace (K :: FldPadExact, m :: RngIntElt, n :: RngIntElt)

-> ModMatFld_FldPadExact

The space of m x n matrices over K.

Creation of matrices

From coefficients

Matrix (K :: FldPadExactElt, nrows :: RngIntElt, ncols :: RngIntElt, cs)

-> ModMatFldElt_FldPadExact

The nrows x ncols matrix over K defined by cs.

Matrix (nrows :: RngIntElt, ncols :: RngIntElt, cs :: [FldPadExactElt])

-> ModMatFldElt_FldPadExact

The nrows x ncols matrix with coefficients cs.

Matrix (cs :: [[FldPadExactElt]])

Matrix (cs :: [ModTupFldElt_FldPadExact])

-> ModMatFldElt_FldPadExact

The matrix whose rows are given by each entry in cs.

Special forms

Zero (M :: ModMatFld_FldPadExact)

ZeroMatrix (M :: ModMatFld_FldPadExact)

ZeroMatrix (K :: FldPadExact, nrows :: RngIntElt, ncols :: RngIntElt)

-> ModMatFldElt_FldPadExact

The zero matrix.

Identity (M :: ModMatFld_FldPadExact)

IdentityMatrix (M :: ModMatFld_FldPadExact)

IdentityMatrix (K :: FldPadExact, nrows :: RngIntElt)

-> ModMatFldElt_FldPadExact

The identity matrix.

ScalarMatrix (M :: ModMatFld_FldPadExact, x)

ScalarMatrix (K :: FldPadExact, nrows :: RngIntElt, x)

ScalarMatrix (nrows :: RngIntElt, x :: FldPadExactElt)

-> ModMatFldElt_FldPadExact

The scalar matrix with x on the diagonal and zero elsewhere.

DiagonalMatrix (M :: ModMatFld_FldPadExact, cs :: [])

DiagonalMatrix (K :: FldPadExact, cs :: [])

DiagonalMatrix (cs :: [FldPadExactElt])

-> ModMatFldElt_FldPadExact

The diagonal matrix with diagonal entries given by cs.

Coercion

The following can be coerced to a matrix in M:

• A matrix in M (or whose components are coercible to the base field)
• A sequence of sequences of components
• A sequence of row vectors
• A sequence of components

IsCoercible (M :: ModMatFld_FldPadExact, X)

-> BoolElt, Any

True if X is coercible to an element of M. If so, also returns the coerced element.

Basic properties of matrix spaces

BaseField (M :: ModMatFld_FldPadExact)

BaseField (m :: ModMatFldElt_FldPadExact)

-> FldPadExact

The base field.

Nrows (M :: ModMatFld_FldPadExact)

Ncols (M :: ModMatFld_FldPadExact)

-> RngIntElt

Number of rows and columns.

Degree (M :: ModMatFld_FldPadExact)

-> RngIntElt

Number of components.

Dimension (M :: ModMatFld_FldPadExact)

-> RngIntElt

The dimension of M.

Generators (M :: ModMatFld_FldPadExact)

-> []

The generators of M.

RowSpace (M :: ModMatFld_FldPadExact)

-> ModTupFld_FldPadExact

The vector space of rows of M.

TransposeSpace (M :: ModMatFld_FldPadExact)

-> ModMatFld_FldPadExact

The space of transposes of elements of M.

Matrix components

Nrows (m :: ModMatFldElt_FldPadExact)

Ncols (m :: ModMatFldElt_FldPadExact)

-> RngIntElt

Number of rows and columns.

Eltseq (m :: ModMatFldElt_FldPadExact)

-> []

The components of m.

Component (m :: ModMatFldElt_FldPadExact, i :: RngIntElt, j :: RngIntElt)

'@' (i :: RngIntElt, j :: RngIntElt, m :: ModMatFldElt_FldPadExact)

-> FldPadExactElt

The jth component of the ith row of m.

Rows (m :: ModMatFldElt_FldPadExact)

-> []

The rows of m.

Row (m :: ModMatFldElt_FldPadExact, i :: RngIntElt)

-> FldPadExactElt

'@' (i :: RngIntElt, m :: ModMatFldElt_FldPadExact)

-> ModTupFldElt_FldPadExact

The ith row of m.

Arithmetic

Addition

'-' (v :: ModTupFldElt_FldPadExact)

'+' (v :: ModTupFldElt_FldPadExact, w :: ModTupFldElt_FldPadExact)

'-' (v :: ModTupFldElt_FldPadExact, w :: ModTupFldElt_FldPadExact)

'&+' (vs :: [ModTupFldElt_FldPadExact])

-> ModTupFldElt_FldPadExact

Negation, addition, subtraction, sum of vectors.

'-' (m :: ModMatFldElt_FldPadExact)

'+' (m :: ModMatFldElt_FldPadExact, n :: ModMatFldElt_FldPadExact)

'-' (m :: ModMatFldElt_FldPadExact, n :: ModMatFldElt_FldPadExact)

'&+' (ms :: [ModMatFldElt_FldPadExact])

-> ModMatFldElt_FldPadExact

Negation, addition, subtraction, sum of matrices.

Scalar multiplication

'*' (v :: ModTupFldElt_FldPadExact, x :: FldPadExactElt)

'*' (x :: FldPadExactElt, v :: ModTupFldElt_FldPadExact)

'/' (v :: ModTupFldElt_FldPadExact, x :: FldPadExactElt)

-> ModTupFldElt_FldPadExact

Scalar multiplication and division of vectors.

Parameters

• Safe := false: (Divide only.) When true, this may be used as an intermediate variable in WithDependencies with the Fast option.

'*' (m :: ModMatFldElt_FldPadExact, x :: FldPadExactElt)

'*' (x :: FldPadExactElt, m :: ModMatFldElt_FldPadExact)

'/' (m :: ModMatFldElt_FldPadExact, x :: FldPadExactElt)

-> ModMatFldElt_FldPadExact

Scalar multiplication and division of matrices.

Parameters

• Safe := false: (Divide only.) When true, this may be used as an intermediate variable in WithDependencies with the Fast option.

Matrix multiplication

InnerProduct (v :: ModTupFldElt_FldPadExact, w :: ModTupFldElt_FldPadExact)

Norm (v :: ModTupFldElt_FldPadExact)

-> ModTupFldElt_FldPadExact

Inner product and norm.

'*' (x :: ModTupFldElt_FldPadExact, y :: ModMatFldElt_FldPadExact)

-> ModTupFldElt_FldPadExact

Vector-matrix multiplication.

'*' (x :: ModMatFldElt_FldPadExact, y :: ModMatFldElt_FldPadExact)

-> ModMatFldElt_FldPadExact

Matrix multiplication.

'^' (x :: ModMatFldElt_FldPadExact, n :: RngIntElt)

-> ModMatFldElt_FldPadExact

Matrix power. Negative powers are allowed for invertible matrices.

Parameters

• Safe := false: When true, this may be used as an intermediate variable in WithDependencies with the Fast option.

Transpose, determinant, inverse

Transpose (m :: ModMatFldElt_FldPadExact)

-> ModMatFldElt_FldPadExact

Transpose.

Determinant (m :: ModMatFldElt_FldPadExact)

-> FldPadExactElt

Determinant.

IsDefinitelyInvertible (m :: ModMatFldElt_FldPadExact)

-> BoolElt, ModMatFldElt_FldPadExact

True if m is definitely invertible. If so, also returns the inverse.

Parameters

• Safe := false: When true, the inverse may be used as an intermediate variable in WithDependencies with the Fast option.