Sage Quick Reference: Linear Algebra
Robert A. Beezer
Sage Version 4.8
http://wiki.sagemath.org/quickref
GNU Free Document License, extend for your own use
Based on work by Peter Jipsen, William Stein

Vector Constructions
Caution: First entry of a vector is numbered 0
u = vector(QQ, [1, 3/2, -1]) length 3 over rationals
v = vector(QQ, {2:4, 95:4, 210:0})
211 entries, nonzero in entry 4 and entry 95, sparse
Vector Operationsu = vector(QQ, [1, 3/2, -1])v = vector(ZZ, [1, 8, -2])
2*u - 3*v linear combination
u.dot_product(v)u.cross_product(v) order: u×v
u.inner_product(v) inner product matrix from parent
u.pairwise_product(v) vector as a result
u.norm() == u.norm(2) Euclidean norm
u.norm(1) sum of entries
u.norm(Infinity) maximum entry
A.gram_schmidt() converts the rows of matrix AMatrix Multiplicationu = vector(QQ, [1,2,3]), v = vector(QQ, [1,2])
A = matrix(QQ, [[1,2,3],[4,5,6]])
B = matrix(QQ, [[1,2],[3,4]])u*A, A*v, B*A, B^6, B^(-3) all possible
B.iterates(v, 6) produces vB 0 , vB 1 , . . . , vB 5rows = False moves v to the right of matrix powers
f(x)=x^2+5*x+3 then f(B) is possible
P∞ 1 kB
B.exp() matrix exponential, i.e. k=0 k!
Matrix Spaces
M = MatrixSpace(QQ, 3, 4) is space of 3 × 4 matrices
A = M([1,2,3,4,5,6,7,8,9,10,11,12])
coerce list to element of M, a 3 × 4 matrix over QQ
M.basis()M.dimension()M.zero_matrix()
Matrix Operations
5*A+2*B linear combination
A.inverse(), A^(-1), ~A, singular is ZeroDivisionError
A.transpose()
A.conjugate() entry-by-entry complex conjugates
A.conjugate_transpose()
A.antitranspose() transpose + reverse orderings
A.adjoint() matrix of cofactors
A.restrict(V) restriction to invariant subspace V

A.pivots() indices of columns spanning column space
A.pivot_rows() indices of rows spanning row space
Pieces of Matrices
Caution: row, column numbering begins at 0A.nrows(), A.ncols()
A[i,j] entry in row i and column j
A[i] row i as immutable Python tuple. Thus,Caution: OK: A[2,3] = 8, Error: A[2][3] = 8
A.row(i) returns row i as Sage vector
A.column(j) returns column j as Sage vector
A.list() returns single Python list, row-major order
A.matrix_from_columns([8,2,8])
new matrix from columns in list, repeats OK
A.matrix_from_rows([2,5,1])
new matrix from rows in list, out-of-order OK
A.matrix_from_rows_and_columns([2,4,2],[3,1])
common to the rows and the columns
A.rows() all rows as a list of tuples
A.columns() all columns as a list of tuples
A.submatrix(i,j,nr,nc)
start at entry (i,j), use nr rows, nc cols
A[2:4,1:7], A[0:8:2,3::-1] Python-style list slicing
Combining Matrices
A.augment(B) A in first columns, matrix B to the rightA.stack(B) A in top rows, B below; B can be a vector
A.block_sum(B) Diagonal, A upper left, B lower rightA.tensor_product(B) Multiples of B, arranged as in AMatrix Constructions
Caution: Row, column numbering begins at 0A = matrix(ZZ, [[1,2],[3,4],[5,6]])
Row Operations
3 × 2 over the integers
Scalar Functions on Matrices
Row Operations: (change matrix in place)
B = matrix(QQ, 2, [1,2,3,4,5,6])
A.rank(), A.right_nullity()
Caution: first row is numbered 0
2 rows from a list, so 2 × 3 over rationals
A.left_nullity() == A.nullity()
A.rescale_row(i,a) a*(row i)
C = matrix(CDF, 2, 2, [[5*I, 4*I], [I, 6]])
A.determinant() == A.det()
A.add_multiple_of_row(i,j,a) a*(row j) + row icomplex entries, 53-bit precision
A.permanent(), A.trace()
A.swap_rows(i,j)
Z = matrix(QQ, 2, 2, 0) zero matrix
A.norm() == A.norm(2) Euclidean norm
Each has a column variant, row→col
D = matrix(QQ, 2, 2, 8)
A.norm(1) largest column sum
For a new matrix, use e.g. B = A.with_rescaled_row(i,a)
diagonal entries all 8, other entries zero
A.norm(Infinity) largest row sum
E = block_matrix([[P,0],[1,R]]), very flexible input
A.norm(’frob’) Frobenius norm
II = identity_matrix(5)5 × 5 identity matrix
Echelon Form√
I = −1, do not overwrite with matrix name
A.rref(), A.echelon_form(), A.echelonize()
Matrix Properties
J = jordan_block(-2,3)Note: rref() promotes matrix to fraction field.is_zero(); .is_symmetric(); .is_hermitian();3 × 3 matrix, −2 on diagonal, 1’s on super-diagonal
A = matrix(ZZ,[[4,2,1],[6,3,2]]).is_square(); .is_orthogonal(); .is_unitary();var(’x y z’); K = matrix(SR, [[x,y+z],[0,x^2*z]]) A.rref() A.echelon_form().is_scalar(); .is_singular(); .is_invertible();symbolic expressions live in the ring SR
12 1 01 2 0.is_one();.is_nilpotent(); .is_diagonalizable()
L = matrix(ZZ, 20, 80, {(5,9):30, (15,77):-6})
0 0 1
0 0 1
20 × 80, two non-zero entries, sparse representation

Eigenvalues and Eigenvectors
Note: Contrast behavior for exact rings (QQ) vs. RDF, CDF
A.charpoly(’t’) no variable specified defaults to x
A.characteristic_polynomial() == A.charpoly()
A.fcp(’t’) factored characteristic polynomial
A.minpoly() the minimum polynomial
A.minimal_polynomial() == A.minpoly()
A.eigenvalues() unsorted list, with mutiplicities
A.eigenvectors_left() vectors on left, _right too
Returns, per eigenvalue, a triple: e: eigenvalue;
V: list of eigenspace basis vectors; n: multiplicity
A.eigenmatrix_right() vectors on right, _left too
Returns pair: D: diagonal matrix with eigenvalues
P: eigenvectors as columns (rows for left version)
with zero columns if matrix not diagonalizable
Eigenspaces: see “Constructing Subspaces”
Decompositions
Note: availability depends on base ring of matrix,
try RDF or CDF for numerical work, QQ for exact“unitary” is “orthogonal” in real caseA.jordan_form(transformation=True)
returns a pair of matrices with: A == P^(-1)*J*P
J: matrix of Jordan blocks for eigenvalues
P: nonsingular matrix
A.smith_form() triple with: D == U*A*V
D: elementary divisors on diagonal
U, V: with unit determinant
A.LU() triple with: P*A == L*U
P: a permutation matrix
L: lower triangular matrix, U: upper triangular matrix
A.QR() pair with: A == Q*R
Q: a unitary matrix, R: upper triangular matrix
A.SVD() triple with: A == U*S*(V-conj-transpose)
U: a unitary matrix
S: zero off the diagonal, dimensions same as A
V: a unitary matrix
A.schur() pair with: A == Q*T*(Q-conj-transpose)
Q: a unitary matrix
T: upper-triangular matrix, maybe 2×2 diagonal blocks
A.rational_form(), aka Frobenius form
A.symplectic_form()
A.hessenberg_form()
A.cholesky() (needs work)

Solutions to Systems
A.solve_right(B) _left too
is solution to A*X = B, where X is a vector or matrix
A = matrix(QQ, [[1,2],[3,4]])b = vector(QQ, [3,4]), then A\b is solution (-2, 5/2)
Vector Spaces
VectorSpace(QQ, 4) dimension 4, rationals as field
VectorSpace(RR, 4) “field” is 53-bit precision reals
VectorSpace(RealField(200), 4)“field” has 200 bit precisionCC^4 4-dimensional, 53-bit precision complexesY = VectorSpace(GF(7), 4) finite
Y.list() has 74 = 2401 vectors
Vector Space Properties
V.dimension()
V.basis()V.echelonized_basis()
V.has_user_basis() with non-canonical basis?
V.is_subspace(W) True if W is a subspace of V
V.is_full() rank equals degree (as module)?
Y = GF(7)^4, T = Y.subspaces(2)T is a generator object for 2-D subspaces of Y
[U for U in T] is list of 2850 2-D subspaces of Y,or use T.next() to step through subspaces
Constructing Subspaces
span([v1,v2,v3], QQ) span of list of vectors over ring
For a matrix A, objects returned arevector spaces when base ring is a fieldmodules when base ring is just a ring
A.left_kernel() == A.kernel() right_ too
A.row_space() == A.row_module()
A.column_space() == A.column_module()
A.eigenspaces_right() vectors on right, _left too
Pairs: eigenvalues with their right eigenspaces
A.eigenspaces_right(format=’galois’)
One eigenspace per irreducible factor of char poly
If V and W are subspaces
V.quotient(W) quotient of V by subspace W
V.intersection(W) intersection of V and W
V.direct_sum(W) direct sum of V and W
V.subspace([v1,v2,v3]) specify basis vectors in a list

Dense versus Sparse
Note: Algorithms may depend on representation
Vectors and matrices have two representations
Dense: lists, and lists of lists
Sparse: Python dictionaries.is_dense(), .is_sparse() to checkA.sparse_matrix() returns sparse version of A
A.dense_rows() returns dense row vectors of A
Some commands have boolean sparse keywordRingsNote: Many algorithms depend on the base ring<object>.base_ring(R) for vectors, matrices,. . .
to determine the ring in use<object>.change_ring(R) for vectors, matrices,. . .
to change to the ring (or field), R
R.is_ring(), R.is_field(), R.is_exact()
Some common Sage rings and fields
ZZ integers, ring
QQ rationals, field
AA, QQbar algebraic number fields, exact
RDF real double field, inexact
CDF complex double field, inexact
RR 53-bit reals, inexact, not same as RDF
RealField(400) 400-bit reals, inexact
CC, ComplexField(400) complexes, too
RIF real interval field
GF(2) mod 2, field, specialized implementations
GF(p) == FiniteField(p) p prime, field
Integers(6) integers mod 6, ring only
CyclotomicField(7) rationals with 7th root of√ unity
QuadraticField(-5, ’x’) rationals with x= −5
SR ring of symbolic expressions
Vector Spaces versus Modules
Module “is” a vector space over a ring, rather than a field
Many commands above apply to modules
Some “vectors” are really module elements
More Help“tab-completion” on partial commands“tab-completion” on <object.> for all relevant methods<command>? for summary and examples<command>?? for complete source code