Python and Linear Algebra: Systems of Linear Equations

I've recently found a need to brush up on my linear algebra (due to some software projects I'm working on). It's been about 10 years since I last did quantum mechanics, where we only used a small subset of liner algebra, and 12 years since an actual linear algebra class. I've used it a bit here and there, but most of it is forgotten.

I thought it might be nice to write up some posts as I relearn some of this material and explore areas of NumPy that I either haven't used before or haven't touched in a while. I anticipate that my approach will be essentially code samples based on problems or examples from various linear algebra texts I own. I want to keep it simple: here's Python, and here's how you solve this linear algebra problem with it.

Solving a System of Linear Equations

Given these equations, find a solution, if one exists:

x - 2y + z = 0
2y - 8z = 8
-4x

• 5y +9z = -9
  
  Keep in mind the geometry of these equations: we havethree equations with three dimensions. Each equation describes a plane. Byseeking a solution that these three all have in common, we are trying to findthe geometric entity defined by all three planes intersecting: a point.
  
  Defining the coefficient and constant matrix separtely, we have:
  
  >>> from numpy import matrix
  >>> a = matrix([
  ... [1, -2,1],
  ... [0, 2, -8],
  ... [-4, 5, 9],
  ... ])
  
  >>> b =matrix([
  ... [0],
  ... [8],
  ... [-9],
  ... ])
  
  Feel free to enter this in whichever way is most comfortable for you :-) I'mjust typing it like this for clear presentation and to maintain a visualconnection to the format often used in text books.
  
  Note that we canwrite the constants matrix "horizontally" and then transpose it:
  
  b= matrix([[0, 8, -9]]).getT()
  
  Now we're going to solve it :-)
  
  >>> from numpy.linalg import solve
  >>> solve(a, b)
  matrix([[29.],
  [ 16.],
  [ 3.]])
  
  To be clear, the solution forthis system of equations is the point located at the coordinates x=29, y=16,and z=3.
  
  What happens if we've got a system of equations whosesolution is inconsistent? Let's give it a try :-)
  
  >>> a = matrix([
  ... [1, -3, 1],
  ... [2, -1, -2],
  ... [1, 2, -3],
  ...])
  >>> b = matrix([[1, 2, -1]]).transpose()
  >>> solve(a, b)
  Traceback (most recent call last):
  File "", line 1, in
  File "/usr/lib/python2.5/site-packages/numpy/linalg/linalg.py",line 235, in solve
  raise LinAlgError, 'Singular matrix'
  numpy.linalg.linalg.LinAlgError: Singular matrix
  
  No solution forthat one!
  
  Network Analysis
  
  -a - b + 0 + 0 + 0 + 20 =0
  0 + 0 - c + d + 0 - 20 = 0
  0 + b + c + 0 + 0 - 20 = 0
  a +0 + 0 + 0 - e + 10 = 0
  0 + 0 + 0 - d + e + 10 = 0
  
  a =matrix([
  [-1, -1, 0, 0, 0],
  [0, 0, -1, 1, 0],
  [0, 1, 1, 0,0],
  [1, 0, 0, 0, -1],
  [0, 0, 0, -1, 1],
  ])
  
  b =matrix([[-20, 20, 20, -10, -10]]).transpose()