Sum of Antidiagonal of a Matrix in R
Last Updated :
18 Oct, 2023
In linear algebra and matrix mathematics, an antidiagonal of a matrix refers to the set of elements that run diagonally from the bottom-left corner to the upper-right corner of the matrix. In R Programming Language Calculating the sum of the antidiagonal elements is a common operation in various data analysis and manipulation tasks using R. This article will provide you with a step-by-step explanation of how to compute the sum of the antidiagonal of a matrix in R.
Concepts Related to the Topic:
- Matrices in R: Understanding how to create and manipulate matrices using R is essential for this task.
- Indexing: Accessing specific elements in a matrix using indexing is crucial for extracting the antidiagonal elements.
Antidiagonal
The antidiagonal of a matrix consists of elements that are positioned along the opposite diagonal, running from the bottom-left to the upper-right of the matrix. and An antidiagonal matrix is a square matrix with all entries zero except those on the diagonal running from the lower left corner to the upper right corner (),
Steps Needed:
Step 1: Create a Matrix
You can start by creating a matrix in R. For this example, let’s create a simple 3×3 matrix.
Step 2: Compute the Antidiagonal Sum
To calculate the sum of the antidiagonal elements, you can use indexing and a loop to iterate through the matrix.
Step 3: Display the Result
You can then display the sum of the antidiagonal elements.
Example1 Sum of Antidiagonal of a Matrix in R
R
matrix_data <- matrix (1:9, nrow = 3)
anti_diag_sum <- 0
for (i in 1: nrow (matrix_data)) {
anti_diag_sum <- anti_diag_sum + matrix_data[i, ncol (matrix_data) - i + 1]
}
matrix_data
cat ( "Sum of the antidiagonal elements:" , anti_diag_sum, "\n" )
|
Output:
[,1] [,2] [,3]
[1,] 1 4 7
[2,] 2 5 8
[3,] 3 6 9
Sum of the antidiagonal elements: 15
Other Approaches
1) Reverse Rows and Sum Diagonal:
R
m <- matrix ( c (2, 3, 1, 4, 2, 5, 1, 3, 7), 3)
print (m)
result1 <- sum ( diag (m[ nrow (m):1, ]))
cat ( "Result 1: Sum of antidiagonal using reversed rows:" , result1, "\n" )
|
Output:
[,1] [,2] [,3]
[1,] 2 4 1
[2,] 3 2 3
[3,] 1 5 7
Result 1: Sum of antidiagonal using reversed rows: 4
First It creates a 3×3 matrix m by using the matrix function. The numbers provided in the c() function are filled into the matrix in column-major order, and the 3 as the second argument specifies that the matrix should have 3 columns. So, m will look like this.
- We reverses the rows of the matrix m and then calculates the sum of the elements along the main diagonal of the reversed matrix. Let’s break it down:
- m[nrow(m):1, ] reverses the rows of matrix m. nrow(m) returns the number of rows (which is 3), and 1 represents the first row. So, m[nrow(m):1, ] rearranges the rows of m in reverse order.
- diag() extracts the diagonal elements from a matrix. In this case, it extracts the diagonal elements from the reversed matrix.
- sum() calculates the sum of the elements in the resulting diagonal.
2) Using Row and Col:
R
m <- matrix ( c (2, 3, 1, 4, 2, 5, 1, 3, 7), 3)
m
result2 <- sum (m[ c ( row (m) + col (m) - nrow (m) == 1)])
cat ( "Result 2: Sum of antidiagonal using row and col:" , result2, "\n" )
|
Output:
[,1] [,2] [,3]
[1,] 2 4 1
[2,] 3 2 3
[3,] 1 5 7
Result 2: Sum of antidiagonal using row and col: 4
- This approach uses row(m) and col(m) to calculate row and column numbers.
- It subtracts nrow(m) to align the diagonal elements with 1.
- sum() calculates the sum of elements that meet the condition.
- The result is 4.
3) Using Sequence of Indexes:
R
m <- matrix ( c (2, 3, 1, 4, 2, 5, 1, 3, 7), 3)
print (m)
n <- nrow (m)
result3 <- sum (m[ seq (n, by = n-1, length = n)])
cat ( "Result 3: Sum of antidiagonal using sequence of indexes:" , result3, "\n" )
|
Output:
[,1] [,2] [,3]
[1,] 2 4 1
[2,] 3 2 3
[3,] 1 5 7
Result 3: Sum of antidiagonal using sequence of indexes: 4
- It uses a sequence of indexes generated by seq(n, by = n-1, length = n) to access the antidiagonal elements.
- sum() calculates the sum of these elements.
- The result is 4.
4) Using Outer Function:
R
m <- matrix ( c (2, 3, 1, 4, 2, 5, 1, 3, 7), 3)
print (m)
n <- nrow (m)
result4 <- sum (m[! c ( outer (1:n, n:1, "-" ))])
cat ( "Result 4: Sum of antidiagonal using outer function:" , result4, "\n" )
|
Output:
[,1] [,2] [,3]
[1,] 2 4 1
[2,] 3 2 3
[3,] 1 5 7
Result 4: Sum of antidiagonal using outer function: 4
- This approach utilizes the outer function to create a logical matrix.
- outer(1:n, n:1, “-“) generates a matrix with values that represent the position relative to the antidiagonal.
- m[!c(outer(1:n, n:1, “-“))] extracts elements where the position is not equal to 0 (on the antidiagonal).
- sum() calculates the sum of these elements.
- This approach can be generalized to other antidiagonals as well.
These methods offer different ways to calculate the sum of antidiagonal elements in a matrix in R, allowing you to choose the one that best suits your specific needs or preferences.
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