An Introduction to Matrix Computations in Python: Just the basics

NumPy is an open source scientific computing package that allows you to use standard mathematical functions and constants like sine or pi. NumPy also allows you to work with arrays of numbers so you can efficiently perform computations.

Note to call a function from an imported package you must use the syntax package.function. To avoid retyping the long name of a package every time, it is standard practice to rename the package as you import.

In addition to the standard trigonometric functions, NumPy also contains \(e^x\) as exp() and natural log as log(), as well as the standard mathematical constants like pi and e.

NumPy arrays can help us efficiently do computations with a collection of numbers all at once. We can also represent vectors or matrices using NumPy arrays. Run the code below that calls the NumPy function array to see how it works.

NumPy also has built in functions to automatically create an array with certain conditions. For example linspace() takes a given interval and sets up an array of evenly spaced numbers on that interval.

Note that the array includes 5, so it takes 11 evenly spaced numbers to get numbers 0.5 apart. Note also that although the number values are evenly spaced, the printed array is not. Namely the default is to space out the entries by the maximum number of decimal places with blank spaces instead of extra zeroes.

Another way to create an array of values for a given interval is arange(), which uses a step value to set up a sequence of numbers within that interval.

Note that the right endpoint of the interval in arange is always excluded (strictly less than).

We can access an entry of a one-dimensional array using it’s index with array_name[index number], where the index numbers the positions of the elements in the array starting with the number 0.

Note: The last element of the array can also be accessed using index -1.

For higher dimensional arrays, you need more than one index to access a single entry array_name[index 1, index 2, index 3]. One index will access a sub-array corresponding to that index.

Slicing selects a range of entries in an array by placing a colon between the desired start and stop index values. Note the entry at the stop index is not included.

Sometimes we want to import an array from a file or spreadsheet. The Python package, Pandas, was created to manipulate spreadsheet data. While the main object in NumPy is a NumPy array, the main object in the Pandas package is a Pandas dataframe, essentially a table with indexed rows and named columns. More information about dataframe (table) manipulation in Pandas will be added to the appendix in future versions of this book.

For now we will just use the Pandas package to import a .csv file as a Pandas dataframe, then convert it into a NumPy array so we can use the computational power of NumPy. Since we don’t have a simple way to upload files into the online textbook, the following code must be used in Google Colab, with the file uploaded temporarily to Colab, or on your local machine with a local version of Python and the appropriate filepath.

Warning: read_csv defaults to reading in a table with headers (column names). Otherwise use pd.read_csv('filepath.csv', header = None)