Matrix Multiplication “*” in R: A Deep Dive

Introduction

As a technical blogger, I’ve encountered numerous questions and conundrums related to matrix multiplication in programming languages. Recently, I came across a Stack Overflow post that caught my attention - the difference between MATLAB’s syntax for matrix multiplication and R’s. In this article, we’ll delve into the intricacies of matrix multiplication in both languages, explore why the syntax differs, and provide practical examples to illustrate key concepts.

Understanding Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves multiplying two matrices together to produce another matrix. The resulting matrix has dimensions determined by the original matrices’ sizes.

Given two matrices A (m x n) and B (n x p), their product C = A * B results in a matrix C of size m x p, where each element c_ij is calculated as:

c_ij = Σ(aIk * b_kj)

where Σ denotes the summation over k from 1 to n.

MATLAB’s Matrix Multiplication Syntax

In MATLAB, matrix multiplication is denoted by the asterisk (*) symbol. This operator performs Hadamard product (element-wise multiplication) if both operands have the same size.

# Example: Hadamard Product in MATLAB
X = [1, 2; 3, 4];
Y = X * X;
Y % prints [15 16; 9 25]

In this example, X is multiplied element-wise by itself, producing a new matrix where each element is the square of its counterpart in X.

R’s Matrix Multiplication Syntax

In R, matrix multiplication is performed using the %*% operator or the [ operator combined with the crossprod() function. The %*% operator performs standard matrix multiplication.

# Example: Standard Matrix Multiplication in R
X <- matrix(c(1, 2, 3, 4), nrow = 2)
tX <- t(X) %*%
X

In this example, tX is the transpose of X, and %*% performs standard matrix multiplication.

Transpose Operation in R

The transpose operation (t()) swaps rows with columns in a matrix. This operator returns a new matrix where each row and column are swapped.

# Example: Transpose Operation in R
X <- matrix(c(1, 2, 3), nrow = 1)
tX <- t(X)
print(tX) % prints [1 2 3]

In this example, the transpose operation swaps the row and column elements of X, producing a new matrix with shape (1 x 3).

Understanding Why %*% Fails in This Case

When attempting to multiply tX by X using the %*% operator, R encounters an error because both matrices do not have compatible dimensions. The resulting matrix has dimensions (5 x 4) and (4 x 3), respectively.

To perform this multiplication, you need to use either crossprod() or the %*% operator with the transpose of one of the matrices.

Conclusion

Matrix multiplication is a fundamental operation in linear algebra that can be challenging to master. Understanding how different programming languages handle matrix multiplication syntax and operations is crucial for effective computation.

In conclusion, the key takeaways from this article are:

• In MATLAB, the asterisk (*) symbol performs Hadamard product (element-wise multiplication) if both operands have the same size.
• In R, matrix multiplication uses the %*% operator or the [ operator combined with crossprod().
• The transpose operation swaps rows with columns in a matrix using the t() function.

By grasping these concepts and syntax differences between MATLAB and R, you’ll be better equipped to tackle complex computational problems involving matrix operations.

Last modified on 2023-09-13