R Academy · Lesson

Eigenvalues and Eigenvectors

Compute eigen decompositions with eigen() and interpret results.

What Are Eigenvalues?

An eigenvector v of matrix A is a non-zero vector that only scales (not rotates) when multiplied by A: Av = λv. The scalar λ is the eigenvalue. They reveal a matrix's intrinsic stretching directions.

# Intuition: A simple scaling matrix
A <- matrix(c(3, 0,
              0, 2), nrow = 2, byrow = TRUE)

# The eigenvectors are the standard basis vectors
# A * c(1,0) = 3 * c(1,0)  -> eigenvalue 3
# A * c(0,1) = 2 * c(0,1)  -> eigenvalue 2

v1 <- c(1, 0)
A %*% v1  # c(3, 0) = 3 * v1

v2 <- c(0, 1)
A %*% v2  # c(0, 2) = 2 * v2

cat('Eigenvalues of a diagonal matrix are its diagonal entries')

eigen(): Computing Eigenvalues

eigen(A) returns a list with $values (eigenvalues sorted by decreasing magnitude) and $vectors (matrix of eigenvectors as columns). The eigenvectors are normalized to unit length.

A <- matrix(c(4, 1,
              2, 3), nrow = 2, byrow = TRUE)

# Compute eigendecomposition
eig <- eigen(A)

# Eigenvalues
eig$values
# [1] 5 2  (descending order)

# Eigenvectors (columns)
eig$vectors
#           [,1]       [,2]
# [1,] 0.7071068 -0.4472136
# [2,] 0.7071068  0.8944272

cat('Each column is one eigenvector (unit length)')

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