Numerical Python Basics05/03/2000
Installing Numerical Python
To use Numerical Python, you'll need to install it first. For the details, see our handy guide with instructions and download locations here.
On its own, Python is a potent tool for mathematics. When extended with the addition of two modules, Numerical Python and the DISLIN data visualization tool, Python becomes a powerhouse for numeric computing and problem solving. Numerical Python (NumPy) extends the Python language to handle matrices and supports the associated mathematics of linear algebra. Like Python, NumPy is a collaborative effort. The principal code writer was Jim Hugunin, a student at MIT. Paul Dubois at Lawrence Livermore National Labs and a few other major supporters (Konrad Hinsen and Travis Oliphant) eventually took on the project.
While NumPy and Python make a good combination, the ability of Python to serve the scientific community is not complete without a data visualization package. An excellent choice for data visualization is the DISLIN package freely provided to the Python community by Helmut Michels of the Max-Planck Institute. DISLIN is an extensive cross platform (Win32/Linux/Unix) library that can be accessed in many languages, including Python. With its support for 2D and 3D plotting and basic drawing commands, you can create informative and eye-catching graphics.
Over the next five months I will show you how to use both NumPy and DISLIN in real applications, starting this week with a review of the useful and interesting world of matrices and linear algebra. You may feel like you are having a flashback to high-school math, but stick with me. It gets interesting.
Introduction to Matrices and Matrix Mathematics
A matrix is a homogenous collection of numbers with a particular shape, like a table of numbers of the same type. The most common matrices are one or two-dimensional. One-dimensional matrices are also called vectors. A, b, and C are examples of matrices; a and b could also be called vectors.
Although a and b both contain the same elements, they are different in shape. The shape of a matrix is described by the number of rows and columns. The a matrix can be described as either a column vector or a 3-by-1 matrix, b is a row-vector or a 1-by-3 matrix. The C matrix is and example of a 3-by-3 matrix.
Addition and Subtraction
You add and subtract matrices, as you might expect, element by element - with only a slight twist. The shape of the matrices must be the same. Thus the following are valid operations,
Matrix multiplication is a more complicated operation. For addition the matrices have to be the same size, but in multiplication only the inner dimensions have to be the same: the columns of the first matrix must be the same as the rows of the second matrix. The resulting matrix will be the size of the remaining outer dimensions. As an example, multiplication of a 1-by-3 matrix by a 3-by-1 matrix results in a 1-by-1 matrix (3 being the 'inner' dimension). If a 5-by-2 matrix is multiplied by a 2-by-5 matrix, a 5-by-5 matrix results.
The FormulaThe governing formula is:
This formula describes how the rows of matrix a and the columns of matrix b can be multiplied to generate matrix c. The indices in c(i,j) are used to specify the element located at the ith row and jth column of the new c matrix. (The same concept holds for the a and b matrices.) The Greek letter Sigma stands for summation. Think of it as a kind of for loop from k=1 to n where n is the size of the inner dimension. Add the resulting value of each trip through the loop together.
The following shows the difference between matrices with the same values but different shapes. The first example is a 2-by-1 matrix multiplied by a 1-by-2 matrix resulting in a 2-by-2 matrix. The second example is a 1-by-2 matrix multiplied by a 2-by-1 matrix resulting in a 1-by-1 matrix (also known as a scalar).
What about division?
Matrix division is undefined; the concept is replaced by the matrix inverse. The inverse of a matrix A is defined by the following equation:
When a matrix and its inverse are multiplied together, the result is the identity matrix or I. The identity matrix is a square matrix where all elements are 0 except along the main diagonal (going from upper left to lower right). A 3-by-3 identity matrix is shown
While the matrix inverse is a bit different from traditional division, the concept follows directly from scalar mathematics where:
There are couple of limitations: the matrix inverse is only defined for square matrices (same number of rows as columns), and it may not exist for certain sets of elements. This is similar to how dividing a scalar by zero is undefined. You just can't work out an inverse for every set of numbers. Where the inverse does exist, it is very useful for solving difficult equations, as we will see later in this tutorial.