Lecture 5: Working with Matrices in R

Overview

• In this lecture, we’ll cover:
  • Introduction to matrices in R.
  • How to create and manipulate matrices for data analysis.
  • Accessing and modifying matrix elements.
  • Practical examples of using matrices for business analytics.

1. Introduction to Matrices

• A matrix is a two-dimensional array in R where all elements must be of the same type (e.g., numeric, character, or logical).
• Matrices are particularly useful for storing and performing mathematical operations on data structured in rows and columns.
• Each element in a matrix can be accessed by specifying its row and column indices.

Example: Creating a Matrix

You can create a matrix using the matrix() function, where you define the elements, number of rows (nrow), and number of columns (ncol).

Example: Creating a Sales Matrix

# Creating a numeric matrix for sales data
sales_matrix <- matrix(c(120, 150, 90, 100, 130, 170, 200, 210, 180), 
                       nrow = 3, 
                       ncol = 3, 
                       byrow = TRUE)
sales_matrix

     [,1] [,2] [,3]
[1,]  120  150   90
[2,]  100  130  170
[3,]  200  210  180

• Explanation: This code creates a matrix representing sales data for three products across three quarters. The argument byrow = TRUE fills the matrix by rows.

Key Matrix Operations:

• Dimensions: Matrices have a specific number of rows and columns. The dim() function returns the dimensions of a matrix.
• Rows and Columns: You can extract specific rows or columns from a matrix for analysis.

dim(sales_matrix)  # Check the dimensions of the matrix (3 rows, 3 columns)

[1] 3 3

2. Accessing Elements in a Matrix

You can access elements within a matrix by specifying their row and column indices using square brackets [row, column].

Example: Accessing an Element in the Matrix

# Accessing the element in the first row, second column
element <- sales_matrix[1, 2]  # Returns 150
element

[1] 150

• Explanation: This code retrieves the element located in the first row and second column of the matrix, which corresponds to the value 150.

Example: Accessing an Entire Row or Column

# Accessing the first row of the matrix
first_row <- sales_matrix[1, ]
first_row

[1] 120 150  90

# Accessing the second column of the matrix
second_column <- sales_matrix[, 2]
second_column

[1] 150 130 210

• Explanation: The first line retrieves the entire first row, while the second line retrieves the entire second column. This is useful for analyzing specific rows or columns in the data.

3. Modifying Matrix Elements

Matrix elements can be updated by assigning new values to specific positions using the same row and column indexing.

Example: Modifying a Matrix Element

# Changing the value of the element in the second row, third column
sales_matrix[2, 3] <- 180
sales_matrix

     [,1] [,2] [,3]
[1,]  120  150   90
[2,]  100  130  180
[3,]  200  210  180

• Explanation: This example updates the value at the second row, third column of the matrix from 170 to 180.

4. Matrix Arithmetic Operations

Matrices support various arithmetic operations such as addition, subtraction, multiplication, and division. These operations are performed element-wise.

Example: Adding Two Matrices

# Creating a second matrix for addition
additional_sales <- matrix(c(10, 15, 5, 10, 20, 15, 10, 5, 20), 
                           nrow = 3, 
                           ncol = 3)

# Adding two matrices
total_sales <- sales_matrix + additional_sales
total_sales

     [,1] [,2] [,3]
[1,]  130  160  100
[2,]  115  150  185
[3,]  205  225  200

• Explanation: This example adds the sales_matrix and additional_sales matrices element-wise, resulting in a new matrix total_sales with the combined sales data.

Example: Matrix Multiplication

# Matrix multiplication (element-wise multiplication)
sales_growth <- sales_matrix * 1.1  # Increase each value by 10%
sales_growth

     [,1] [,2] [,3]
[1,]  132  165   99
[2,]  110  143  198
[3,]  220  231  198

• Explanation: This example multiplies each element in the sales_matrix by 1.1, increasing sales by 10%. Matrix multiplication is useful when applying transformations or adjustments across all elements.

5. Practical Applications of Matrices in Business Analytics

Matrices are often used in business analytics to represent: - Sales data: Organizing data for multiple products across different time periods. - Cost and revenue analysis: Using matrices to compute profits by subtracting costs from revenue data. - Forecasting: Applying growth rates or trends to matrices for future projections.

Example: Computing Profit from Sales and Cost Matrices

# Defining a cost matrix
cost_matrix <- matrix(c(70, 90, 50, 60, 80, 120, 150, 160, 140), 
                      nrow = 3, 
                      ncol = 3)

# Calculating profit by subtracting cost from sales
profit_matrix <- sales_matrix - cost_matrix
profit_matrix

     [,1] [,2] [,3]
[1,]   50   90  -60
[2,]   10   50   20
[3,]  150   90   40

• Explanation: This code calculates profit by subtracting a cost matrix from the sales matrix, resulting in a matrix of profits for each product across different periods.

Key Takeaways

• Matrices are two-dimensional arrays that hold elements of the same type and are useful for performing mathematical operations across rows and columns.
• You can create, access, and modify matrix elements in R using indexing.
• Matrix arithmetic (e.g., addition and multiplication) allows for efficient calculations, especially when working with datasets like sales or revenue.

Looking Forward

• In the next lecture, we’ll dive into working with data frames, one of the most common and flexible data structures in R, allowing you to handle datasets with different types of data in each column.