Linear Algebra Reference Sheet printable pdf download

Sage Quick Reference: Linear Algebra

indices of columns spanning column space

A.pivots()

Robert A. Beezer

indices of rows spanning row space

A.pivot_rows()

Matrix Multiplication

Sage Version 4.8

u = vector(QQ,

[1,2,3]),

v = vector(QQ, [1,2])

Pieces of Matrices

A = matrix(QQ, [[1,2,3],[4,5,6]])

GNU Free Document License, extend for your own use

Caution: row, column numbering begins at 0

B = matrix(QQ, [[1,2],[3,4]])

Based on work by Peter Jipsen, William Stein

all possible

u*A, A*v, B*A, B^6,

B^(-3)

A.nrows(),

A.ncols()

produces

entry in row i and column j

B.iterates(v, 6)

A[i,j]

rows = False moves v to the right of matrix powers

row i as immutable Python tuple. Thus,

A[i]

Vector Constructions

then

is possible

Caution: OK: A[2,3] = 8, Error: A[2][3] = 8

f(x)=x^2+5*x+3

f(B)

Caution: First entry of a vector is numbered 0

matrix exponential, i.e.

returns row i as Sage vector

B.exp()

A.row(i)

length 3 over rationals

u = vector(QQ, [1, 3/2, -1])

returns column j as Sage vector

A.column(j)

v = vector(QQ, {2:4, 95:4, 210:0})

returns single Python list, row-major order

A.list()

Matrix Spaces

211 entries, nonzero in entry 4 and entry 95, sparse

A.matrix_from_columns([8,2,8])

is space of 3

4 matrices

M = MatrixSpace(QQ, 3, 4)

new matrix from columns in list, repeats OK

A = M([1,2,3,4,5,6,7,8,9,10,11,12])

Vector Operations

A.matrix_from_rows([2,5,1])

coerce list to element of M, a 3

4 matrix over QQ

new matrix from rows in list, out-of-order OK

u = vector(QQ, [1, 3/2, -1])

M.basis()

A.matrix_from_rows_and_columns([2,4,2],[3,1])

v = vector(ZZ, [1, 8, -2])

M.dimension()

common to the rows and the columns

linear combination

2*u - 3*v

M.zero_matrix()

all rows as a list of tuples

u.dot_product(v)

A.rows()

all columns as a list of tuples

order: u v

A.columns()

u.cross_product(v)

Matrix Operations

inner product matrix from parent

A.submatrix(i,j,nr,nc)

u.inner_product(v)

vector as a result

linear combination

start at entry (i,j), use nr rows, nc cols

u.pairwise_product(v)

5*A+2*B

A.inverse(), A^(-1), ~A, singular is ZeroDivisionError

Python-style list slicing

Euclidean norm

A[2:4,1:7],

A[0:8:2,3::-1]

u.norm() == u.norm(2)

sum of entries

u.norm(1)

A.transpose()

entry-by-entry complex conjugates

u.norm(Infinity)

maximum entry

A.conjugate()

Combining Matrices

converts the rows of matrix A

A.conjugate_transpose()

A.gram_schmidt()

A in ﬁrst columns, matrix B to the right

A.augment(B)

transpose + reverse orderings

A.antitranspose()

A in top rows, B below; B can be a vector

A.stack(B)

matrix of cofactors

A.adjoint()

Matrix Constructions

Diagonal, A upper left, B lower right

A.block_sum(B)

restriction to invariant subspace V

A.restrict(V)

Multiples of B, arranged as in A

Caution: Row, column numbering begins at 0

A.tensor_product(B)

A = matrix(ZZ, [[1,2],[3,4],[5,6]])

Row Operations

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()

2 rows from a list, so 2

3 over rationals

Caution: ﬁrst row is numbered 0

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*(row j) + row i

complex entries, 53-bit precision

A.add_multiple_of_row(i,j,a)

A.permanent(),

A.trace()

zero matrix

Z = matrix(QQ, 2, 2, 0)

A.swap_rows(i,j)

Euclidean norm

A.norm() == A.norm(2)

Each has a column variant, row col

D = matrix(QQ, 2, 2, 8)

largest column sum

A.norm(1)

For a new matrix, use e.g.

diagonal entries all 8, other entries zero

B = A.with_rescaled_row(i,a)

largest row sum

A.norm(Infinity)

block_matrix([[P,0],[1,R]]), very ﬂexible input

E =

Frobenius norm

A.norm(’frob’)

5 identity matrix

II = identity_matrix(5)

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

3 matrix,

2 on diagonal, 1’s on super-diagonal

.is_zero(); .is_symmetric(); .is_hermitian();

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()

symbolic expressions live in the ring SR

.is_scalar(); .is_singular(); .is_invertible();

2 1 0

.is_one(); .is_nilpotent();

.is_diagonalizable()

L = matrix(ZZ, 20, 80, {(5,9):30, (15,77):-6})

0 0 1

0 1

80, two non-zero entries, sparse representation

Linear Algebra Reference Sheet

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