Rank of the Matrix in R Programming Language
The rank of matrix is used in linear algebra to describe the maximum number of linearly independent rows or columns of the matrix. The rank of matrix is written as rank(A) for a matrix A. Rank of matrix is a basic concept of linear algebra which provides certain useful information about properties and behavior of matrix. It is trying to measure up the “dimensionality” or number of pieces in information that are actually independent.
- Row Rank & Column Rank: The row rank of a matrix is the maximum number of linearly independent rows. The column rank of a matrix is the maximum number of linearly independent columns.
- Rank and Nullity Theorem: According to this theorem, the number of columns in any matrix A is equal to the sum of its rank and nullity (dimension of the null space).
- Reduced Row Echelon Form (RREF): The rank of a matrix is the number of non-zero rows in its reduced row echelon form (RREF). It is obtained through row operations that simplify the matrix.
- Relation to Solutions of Linear Systems: In a system of linear equations, the rank of a coefficient matrix corresponds to the number of solutions in the system. There is only one solution for the system if the rank is the same as the number of variables. The system has an infinite number of solutions if the rank is smaller than the number of variables.
- Linearly Independent: If a set of vectors cannot be described as a linear combination of one another,, then they are termed linearly independent.
For an m x n matrix:
- The rank is at most m or n, whichever is smaller.
- For a matrix to have full row rank, the rank should be equal to m.
- Similarly,, for a full column rank,, the rank should be equal to n.
- If rank(A) < min(m, n), the matrix is said to be rank-deficient.
Rank of a matrix is of great significance
- The rank of the matrix determines the number of independent equations.
- For a matrix to be invertible, it should have full rank.
- Rank is quite useful in data analysis and machine learning; it is used to determine the number of independent features or variables.
How to find the rank of a matrix in R
A matrix is an arrangement of data in a row- and column-wise fashion or a matrix is nothing but a set of particular kinds of data like numbers. It has many applications in mathematics, Physics, engineering, etc. The number of rows and columns defines the matrix size called an order. For example, if the matrix has m rows and n columns, the order of the matrix would be m x n. A matrix in R Programming Language is a 2D array that can store numeric, character, or logical data.
# Create a matrix of order 2 x 3 in R
my_matrix <- matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, ncol = 3, byrow = TRUE)
# Print the matrix
print(my_matrix)
Output:
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
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