Linear algebra

Monomorphic arrays

Module la provides an array data type which can be used to calculate with coordinate vectors and matrices.

use la: vector, matrix

v = vector(1,2)

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

print(A*v)

Polymorphic arrays

use math.la: matrix

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

print(A^40)

# Output (long integers instead of floating point numbers):
# matrix(
#   [38418114959269691024862069751, 55991602170538933080248818850],
#   [83987403255808399620373228275, 122405518215078090645235298026]
# )

It is possible to combine this with rational numbers. A rational number a/b is denoted as rat(a,b).

use math.la: matrix
use math.rational: rat

A = matrix(
  [rat(4,1),rat(2,3)],
  [rat(9,5),rat(1,2)]
)

print(A^2)

# Output:
# matrix(
#   [86/5, 3],
#   [81/10, 29/20]
# )

Vector algebra

Basic operations are stated directly.

use math.la: vector

a = vector(1,2)
b = vector(3,4)

# Linear combination
c = 2*a+4*b

# Scalar product
s = a*b

# Absolute value
r = abs(a)

# Unit vector
e = a/abs(a)

The orthogonal projection of v onto a line of direction u is given by:

P = |u,v| (u*v)/(u*u)*u

Matrix algebra

Basic operations are stated directly.

use math.la: vector, matrix, scalar
use math: pi, sin, cos

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

rot = |phi| matrix(
  [cos(phi),-sin(phi)],
  [sin(phi),cos(phi)]
)

deg = pi/180
B = rot(90*deg)

# Multiplication matrix*vector
v = vector(1,2)
w = A*v

# Multiplication matrix*matrix
C = A*B

# Matrix power
C = A^2

# Transposition
C = A.T

# Trace
t = A.tr

# n×n identity matrix
E = scalar(n,1,0)

Column and row vectors

Column and row matrices may be used as coordinate vectors.

use math.la: matrix

# A row vector
v = matrix([1,2])

# A col vector
w = matrix([3,4]).T

# Scalar product
s = v*w

# Outer product
M = w*v

Complex numbers

Vectors and matrices of complex numbers are handled almost like the real versions, with the difference that scalar products need an explicit conjugation operation.

use math.la: vector, matrix

v = vector(1+2i,3+4i)
w = vector(5+6i,7+8i)

# Scalar product
s = v.conj*w

# Absolute value
r = abs(v)

A = matrix(
  [1+2i, 3+4i]
  [5+6i, 7+8i]
)

# Conjugate matrix
B = A.conj

# Conjugate transpose
B = A.T.conj
B = A.H

# Bra-ket notation
ketv = matrix([1+2i,3+4i]).T
ketw = matrix([5+6i,7+8i]).T
brav = ketv.H
braw = ketw.H

# Scalar product
s = brav*ketw

# Absolute value
# (uses Frobenius norm for compability with vectors)
r = abs(ketv)

Matrix inversion

There are functions, providing computation of determinant and inverse of a matrix.

use math.la: matrix
use math.la.inversion: det, inv

A = matrix(
  [1,2,3],
  [4,5,6],
  [7,7,9]
)

print(det(A))
print(inv(A))

Type conversion

Often one needs to convert between different representations.

use math.la: vector, matrix, array

# Turning a list into a vector
v = vector(*a)

# Turning a vector into a list
a = v.list()

# Turning a ragged array into a matrix
A = matrix(*a)

Indexing

Vectors are indexed like lists.

v = vector(1,2)
print(v[0])

Sometimes, rather than indexing a vector, the elements should be unpacked into named coordinates.

# Unpacking an indexed object of unknown type
[x,y] = v

# Unpacking a function's argument
f = |[x,y]| x^2+y^2
print(f(vector(1,2)))