How to Create Matrices in R

Matrices are two-dimensional data structures in R with rows and columns. They’re essential for mathematical operations and data analysis.

Creating Matrices

Using matrix()

# Basic matrix
m <- matrix(1:6, nrow = 2, ncol = 3)
print(m)
#      [,1] [,2] [,3]
# [1,]    1    3    5
# [2,]    2    4    6

# Fill by row instead of column
m <- matrix(1:6, nrow = 2, ncol = 3, byrow = TRUE)
print(m)
#      [,1] [,2] [,3]
# [1,]    1    2    3
# [2,]    4    5    6

With Row and Column Names

m <- matrix(1:6, nrow = 2, ncol = 3,
            dimnames = list(c("Row1", "Row2"), 
                          c("Col1", "Col2", "Col3")))
print(m)
#      Col1 Col2 Col3
# Row1    1    3    5
# Row2    2    4    6

From Vectors

# Combine vectors as rows
v1 <- c(1, 2, 3)
v2 <- c(4, 5, 6)
m <- rbind(v1, v2)  # Row bind
print(m)

# Combine vectors as columns
m <- cbind(v1, v2)  # Column bind
print(m)

Accessing Elements

Single Elements

m <- matrix(1:9, nrow = 3, ncol = 3)

# Single element [row, column]
print(m[1, 2])  # Row 1, Column 2

# Entire row
print(m[1, ])   # Row 1

# Entire column
print(m[, 2])   # Column 2

Multiple Elements

m <- matrix(1:9, nrow = 3, ncol = 3)

# Multiple rows
print(m[1:2, ])

# Multiple columns
print(m[, 2:3])

# Submatrix
print(m[1:2, 2:3])

# By row/column names
m <- matrix(1:9, nrow = 3, 
            dimnames = list(c("A", "B", "C"), c("X", "Y", "Z")))
print(m["A", "Y"])

Matrix Operations

Basic Arithmetic

m1 <- matrix(1:4, nrow = 2)
m2 <- matrix(5:8, nrow = 2)

# Element-wise operations
print(m1 + m2)  # Addition
print(m1 - m2)  # Subtraction
print(m1 * m2)  # Element-wise multiplication
print(m1 / m2)  # Element-wise division

# Scalar operations
print(m1 * 2)   # Multiply all by 2
print(m1 + 10)  # Add 10 to all

Matrix Multiplication

m1 <- matrix(1:4, nrow = 2)  # 2x2
m2 <- matrix(1:6, nrow = 2)  # 2x3

# Matrix multiplication
result <- m1 %*% m2
print(result)  # 2x3 result

Transpose

m <- matrix(1:6, nrow = 2, ncol = 3)
print(m)
#      [,1] [,2] [,3]
# [1,]    1    3    5
# [2,]    2    4    6

t_m <- t(m)
print(t_m)
#      [,1] [,2]
# [1,]    1    2
# [2,]    3    4
# [3,]    5    6

Matrix Functions

Dimensions

m <- matrix(1:12, nrow = 3, ncol = 4)

dim(m)    # Dimensions: 3 4
nrow(m)   # Number of rows: 3
ncol(m)   # Number of columns: 4
length(m) # Total elements: 12

Summary Statistics

m <- matrix(1:9, nrow = 3)

sum(m)      # Sum of all elements
mean(m)     # Mean of all elements
min(m)      # Minimum
max(m)      # Maximum

# By row or column
rowSums(m)   # Sum of each row
colSums(m)   # Sum of each column
rowMeans(m)  # Mean of each row
colMeans(m)  # Mean of each column

Apply Function

m <- matrix(1:9, nrow = 3)

# Apply function to rows (MARGIN = 1)
apply(m, 1, sum)  # Same as rowSums

# Apply function to columns (MARGIN = 2)
apply(m, 2, mean)  # Same as colMeans

# Custom function
apply(m, 1, function(x) max(x) - min(x))  # Range per row

Linear Algebra

Determinant and Inverse

m <- matrix(c(4, 2, 7, 6), nrow = 2)

# Determinant
det(m)

# Inverse
solve(m)

# Verify: m * inverse = identity
m %*% solve(m)

Solving Linear Equations

# Solve Ax = b
A <- matrix(c(3, 1, 2, 4), nrow = 2)
b <- c(5, 11)

x <- solve(A, b)
print(x)

# Verify
A %*% x  # Should equal b

Eigenvalues and Eigenvectors

m <- matrix(c(4, 1, 2, 3), nrow = 2)

eigen_result <- eigen(m)
print(eigen_result$values)   # Eigenvalues
print(eigen_result$vectors)  # Eigenvectors

Special Matrices

# Identity matrix
diag(3)
#      [,1] [,2] [,3]
# [1,]    1    0    0
# [2,]    0    1    0
# [3,]    0    0    1

# Diagonal matrix
diag(c(1, 2, 3))

# Extract diagonal
m <- matrix(1:9, nrow = 3)
diag(m)  # c(1, 5, 9)

# Matrix of zeros
matrix(0, nrow = 3, ncol = 3)

# Matrix of ones
matrix(1, nrow = 2, ncol = 4)

Practical Examples

Correlation Matrix

# Create sample data
data <- matrix(rnorm(30), ncol = 3)
colnames(data) <- c("X", "Y", "Z")

# Correlation matrix
cor_matrix <- cor(data)
print(round(cor_matrix, 2))

Transformation Matrix

# 2D rotation matrix
rotate <- function(theta) {
    matrix(c(cos(theta), sin(theta), 
            -sin(theta), cos(theta)), 
           nrow = 2)
}

# Rotate point (1, 0) by 90 degrees
point <- c(1, 0)
rotated <- rotate(pi/2) %*% point
print(round(rotated, 2))  # (0, 1)

Distance Matrix

# Points
points <- matrix(c(0, 0, 1, 0, 0, 1, 1, 1), 
                 nrow = 4, byrow = TRUE)

# Euclidean distance matrix
dist(points)

Summary

• Create with matrix(), rbind(), or cbind()
• Access elements with [row, column]
• %*% for matrix multiplication, * for element-wise
• Use t() for transpose, solve() for inverse
• apply() to apply functions across rows or columns
• rowSums(), colSums(), rowMeans(), colMeans() for aggregation