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November 23rd









November 23rd

Linear Algebra

Vector Constructions
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, non zero in entry 4 & entry 95, sparse

Linear Combinations
u = vector(QQ, [1, 3/2, -1])
v = vector(ZZ, [1, 8, -2]
2*u - 3*v is (-1, -22, 4)

Vector Operations
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() Euclidean norm
u.norm(1) sum of entries
u.norm(Infinity) maximum entry
A.gram_schmidt() apply to vectors that are rows of ma-
trix A

Matrix Constructions
Row and column numbering begins at 0!
A = matrix(ZZ, [[1,2],[3,4],[5,6]])
3 × 2 over the integers
B = matrix(QQ, 2, [1,2,3,4,5,6])
2 rows from a list, so 2 × 3 over rationals
C = matrix(CDF, 2, 2, [[5*I, 4*I], [I, 6]])
complex entries , 53-bit precision
Z = matrix(QQ, 2, 2, 0) zero matrix
D = matrix(QQ, 2, 2, 8)
diagonal entries all 8, other entries zero
I = identity_matrix(5) 5 × 5 identity matrix
J = jordan_block(-2,3)
3 × 3 matrix, -2 on diagonal, 1's on super-diagonal

Matrix Multiplication
u = 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 all possible
B.iterates(v, 6) produces
f(x)=x^2+5*x+3 then f(B) is possible
B.exp() matrix exponential ,

Matrix Spaces
M = MatrixSpace(QQ, 3, 4)
dimension 12 space of 3 × 4 matrices
A = M([1,2,3,4,5,6,7,8,9,10,11,12])
is a 3 × 4 matrix, an element of M
M.basis()
M.dimension()
M.zero_matrix()

Matrix Operations
5*A scalar multiplication
A.inverse, also A^(-1), ~A
Zero DivisionError if singular
A.transpose()
A.antitranspose() transpose + reverse order
A.adjoint() matrix of cofactors
A.conjugate() entry-by-entry complex conjugates
A.restrict(V) restriction on invariant subspace V

Row Operations
Row Operations: ( change matrix in place)
Recall: rst row is numbered 0
A.rescale_row(i,a) a*(row i)
A.add_multiple_of_row(i,j,a) a*(row j) + row i
A.swap_rows(i,j)
Each has a column variant, row!col
For a new matrix, use e.g. B = A.with_rescaled_row(i,a)

Echelon Form
A.echelon_form(), A.echelonize(), A.hermite_form()
Careful: Base ring a ects results!
A = matrix(ZZ,[[4,2,1],[6,3,2]])
B = matrix(QQ,[[4,2,1],[6,3,2]])
A .echelon_form() B.echelon_form()

A.pivots() indices of columns spanning column space
A.pivot_rows() indices of rows spanning row space
(These do not require matrix to be in echelon form)

Pieces of Matrices
Recall: row and column numbering begins at 0
A.nrows()
A.ncols()
A[i,j] entry in row i and column j
A[i] row i as Python tuple
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

Combining Matrices
A.augment(B) A in rst columns, B to the right
A.stack(B) A in top rows, B below
A.block_sum(B) Diagonal, A upper left, B lower right
A.tensor_product(B) Multiples of B, arranged as in A

Scalar Functions on Matrices
A.rank()
A.nullity() == A.left_nullity()
A.right_nullity()
A.determinant() == A.det()
A.permanent()
A.trace()
A.norm() == A.norm(2) Euclidean norm
A.norm(1) largest column sum
A.norm(Infinity) largest row sum
A.norm('frob') Frobenius norm

Matrix Properties
.is_zero() (totally?), .is_one() (identity matrix?),
.is_scalar() (multiple of identity?), .is_square(),
.is_symmetric(), .is_invertible(), .is_nilpotent()

Eigenvalues
A.charpoly('t') no variable speci ed gets 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() there is a _right version too
Returns a list of triples, one per eigenvalue:
lambda: the eigenvalue
V: list of vectors, basis for eigenspace
n: algebraic multiplicity
A.eigenmatrix_right() there is a _left version too
Returns two matrices :
D: diagonal matrix with eigenvalues
P: eigenvectors as columns (rows for left version)

Decompositions
A.jordan_form(transformation=True)
returns a pair of matrices:
J: matrix of Jordan blocks for eigenvalues
P: nonsingular matrix
so A == P^(-1)*J*P

A.smith_form() returns a triple of matrices:
D: elementary divisors on diagonal
U: with unit determinant
V: with unit determinant
so D == U*A*V

A.LU() returns a triple of matrices:
P: a permutation matrix
L: lower triangular matrix
U: upper triangular matrix
so P*A == L*U

A.QR() returns a pair of matrices:
Q: an orthogonal matrix
R: upper triangular matrix
so A == Q*R

A.SVD() returns a triple of matrices:
U: an orthogonal matrix
S: zero o the diagonal, same dimensions as A
V: an orthogonal matrix
so A == U*S*V', with V' = V-conjugate-transpose

A.symplectic_form()
A.hessenberg_form()
A.cholesky()

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 returns the solution (-2, 5/2)

Vector Spaces
U = VectorSpace(QQ, 4) dimension 4, rationals as eld
V = VectorSpace(RR, 4) \ eld" is 53-bit precision reals
W = VectorSpace(RealField(200), 4)
\ eld" has 200 bit precision
X = CC^4 4-dimensional, 53-bit precision complexes
Y = VectorSpace(GF(7), 4) nite
Y.finite() returns True
len(Y.list()) returns 74 = 2401 elements

Vector Space Properties
V.dimension()
V.basis()
V.echelonized_basis()
V.has_user_basis() with non-canonical basis?
V.is_subspace(W) True if
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

Constructing Subspaces
span([v1,v2,v3], QQ) span of list of vectors over ring

For a matrix A, objects returned are
vector spaces when base ring is a eld
modules 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() _left too
Pairs: eigen value with right eigenspace

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
Vectors and matrices have two representations
Dense: lists, and lists of lists
Sparse: Python dictionaries
.is_dense(), .is_sparse() to check
A.sparse_matrix() returns sparse version of A
A.dense_rows() returns dense row vectors of A
Some commands have boolean sparse keyword

Rings
Many linear algebra algorithms depend on the base ring
.base_ring(R) for vectors, matrices,. . .
to determine the ring in use
.change_ring(R), for vectors, matrices,. . .
to change to the ring (or eld), R,
R.is_ring(), R.is_field()
R.is_integral_domain(), R.is_exact()

Some ring and elds
ZZ integers, ring
QQ rationals, eld
QQbar algebraic eld, exact
RDF real double eld, inexact
RR 53-bit reals, inexact
RealField(400) 400-bit reals, inexact
CDF, CC, ComplexField(400) complexes, too
RIF real interval eld
GF(2) mod 2, eld, special case
GF(p) p prime, eld
Integers(6) integers mod 6, ring
CyclotomicField(7) rationals with 7th root of unity
QuadraticField(-5, 'x') rationals adjoin

Vector Spaces versus Modules
A module is \ like " a vector space over a ring, not a eld
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

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