Term$-$by$-$Document Matrix

A 'term$-$by$-$document' matrix is a matrix that relates search terms to documents that contain

relevant search terms. Consider a simple example with nine search terms and seven document titles

For this example, a $97$ term$-$by$-$document matrix can be constructed as

$A=\left[\begin{array}{ccccccc}0& 1& 0& 1& 1& 0& 1\\ 0& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 1& 1& 0& 0& 0& 0\\ 1& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0\\ 0& 0& 1& 1& 0& 0& 0\\ 1& 0& 0& 1& 0& 0& 0\end{array}\right]$

where the rows index indicates the term number and the column index indicates the document

number. An entry of one indicates that the term indexed by the row can be found in the document

title indexed by the column. For example, row $5$ indicates that T$5$ can be found in documents D$2$

and D$3.$ A given column in the matrix indicates which terms are contained within the associated

document title. For example, column $3$ indicates that terms T$2,$ T$5$ and T$8$ can be found in

document title D$3.$

Search Query

Consider searching the above database represented by the term$-$by$-$document matrix for books on

'child proofing'. In this case we would search using a term query vector

$q=\left[\begin{array}{ccccccccc}0& 1& 0& 0& 0& 0& 1& 0& 0\end{array}\right]$

which shows that terms T$2$ and T$7$ must be found in the desired document title. The goal then

would be to 'compare' the query vector against all column vectors in the term$-$by$-$document matrix

and order the comparison results with some kind of rating system.

Given the term$-$by$-$document matrix A and the query vector q shown in the previous sections above,

use Python and Numpy to perform the following computations.

$1.$ Construct the matrix A and the query vector q$.$

$2.$ Construct a new matrix A by normalizing the columns of A so that each column in A has

unit norm.

$3.$ Normalize the query vector q to form a unit vector q$.$

$4.$ Compute the dot product between q and every column vector of A$.$ Interpret your calculation

and determine which document titles are most related to the query.

$5.$ Upload your well$-$documented Python script $.$py le to the Search query matrix application

Assignment in Canvas. Make sure to create a comments section that provides the

results of your computation and explains your results within your Python script

or you will not receive credit