R Academy · Lesson

Eigenvalues and Eigenvectors

Compute eigen decompositions with eigen() and interpret results.

Lesson 3 of 413 steps

Eigenvalues and Eigenvectors is a free R Academy lesson on CoddyKit. This is lesson 3 of 4. Any 3 lessons of this learning path are free to read in full — after that, CoddyKit PRO unlocks every lesson, plus hands-on practice with a built-in code editor and a 24/7 AI tutor. It is part of the R Academy learning path, and your progress syncs across the web and the CoddyKit app. The R Academy course includes 4 lessons in total.

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)')

Verifying Av = lambda * v

To verify, check that A %*% v - lambda * v is essentially zero. Due to floating point, use max(abs(...)) and check it's near machine epsilon rather than testing exact equality.

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

lambda1 <- eig$values[1]   # 5
v1      <- eig$vectors[, 1]

# Verify: Av - lambda*v should be ~0
residual <- A %*% v1 - lambda1 * v1
max(abs(residual))  # ~ 1e-16

# Check all eigenpairs at once
# A V = V diag(lambda) where V = eigenvectors matrix
V <- eig$vectors
Lambda <- diag(eig$values)
err <- A %*% V - V %*% Lambda
max(abs(err))  # near zero

Symmetric Matrices: Real Eigenvalues

Symmetric matrices (A = t(A)) always have real eigenvalues and orthogonal eigenvectors. This is the spectral theorem — essential in statistics (covariance matrices are symmetric positive semi-definite).

# Symmetric matrix
S <- matrix(c(4, 2, 2,
              2, 3, 1,
              2, 1, 5), nrow = 3, byrow = TRUE)

# All symmetric: A = t(A)
all.equal(S, t(S))  # TRUE

# Eigenvalues are real
eig <- eigen(S)
eig$values  # All real numbers
Im(eig$values)  # All zero

# Eigenvectors are orthogonal: V'V = I
V <- eig$vectors
round(t(V) %*% V, 10)  # Identity matrix
cat('Symmetric -> real eigenvalues, orthogonal eigenvectors')

Asymmetric Matrices: Complex Eigenvalues

Non-symmetric matrices can have complex eigenvalues (complex conjugate pairs). R returns them as complex numbers. The real part governs growth/decay; the imaginary part governs rotation/oscillation.

# Rotation-like matrix -> complex eigenvalues
A <- matrix(c(0, -1,
              1,  0), nrow = 2, byrow = TRUE)
# This is a 90-degree rotation matrix
eig <- eigen(A)
eig$values
# [1] 0+1i 0-1i  (pure imaginary!)

# Real part: zero -> neither grows nor decays
Re(eig$values)  # 0 0

# Imaginary part: indicates rotation
Im(eig$values)  # 1 -1

# Another example: stable spiral
B <- matrix(c(-1, -2,
               2, -1), nrow = 2, byrow = TRUE)
eigen(B)$values  # -1 +/- 2i

Eigenvalue Decomposition A = VLV^-1

A diagonalizable matrix A can be written as A = V Λ V⁻¹, where V is the eigenvector matrix and Λ = diag(eigenvalues). This decomposition unlocks matrix powers: A^n = V Λ^n V⁻¹.

A <- matrix(c(4, 1,
              2, 3), nrow = 2, byrow = TRUE)
eig <- eigen(A)
V      <- eig$vectors
Lambda <- diag(eig$values)

# Reconstruct A = V %*% Lambda %*% solve(V)
A_reconstructed <- V %*% Lambda %*% solve(V)
all.equal(A, A_reconstructed)  # TRUE

# Matrix power A^3 using eigendecomposition
A_cubed_eig <- V %*% diag(eig$values^3) %*% solve(V)
A_cubed_direct <- A %*% A %*% A
all.equal(Re(A_cubed_eig), A_cubed_direct)  # TRUE

PCA Concept via Covariance Eigenvalues

Principal Component Analysis (PCA) uses eigendecomposition of the covariance matrix. Eigenvectors give the directions of maximum variance (principal components); eigenvalues give the variance along each direction.

set.seed(42)
# Correlated 2D data
x1 <- rnorm(100)
x2 <- 0.8 * x1 + 0.6 * rnorm(100)
X <- cbind(x1, x2)

# Covariance matrix
C <- cov(X)
print(round(C, 3))

# Eigendecomposition of covariance matrix
eig <- eigen(C)
cat('Eigenvalues (variance explained):\n')
print(eig$values)
cat('PC1 direction:\n')
print(eig$vectors[, 1])

# Variance explained by PC1
prop_var <- eig$values[1] / sum(eig$values)
cat('PC1 explains:', round(100 * prop_var, 1), '%')

Spectral Radius

The spectral radius ρ(A) = max|λᵢ| is the largest absolute eigenvalue. It determines stability in iterative algorithms: if ρ < 1, iterations converge; if ρ > 1, they diverge.

spectral_radius <- function(A) {
  max(Mod(eigen(A)$values))
}

# Convergent matrix: spectral radius < 1
A_conv <- matrix(c(0.5, 0.2,
                   0.1, 0.3), nrow = 2, byrow = TRUE)
spectral_radius(A_conv)  # < 1 -> iterations converge

# Divergent matrix: spectral radius > 1
A_div <- matrix(c(2, 0.5,
                  0.3, 1.5), nrow = 2, byrow = TRUE)
spectral_radius(A_div)  # > 1 -> iterations diverge

# For positive definite A, spectral radius = max eigenvalue
cat('Spectral radius determines iterative stability')

Determinant and Trace via Eigenvalues

The determinant equals the product of eigenvalues; the trace equals their sum. These relationships connect algebraic properties to eigenvalues and provide sanity checks for your eigen() results.

A <- matrix(c(5, 2,
              1, 4), nrow = 2, byrow = TRUE)
eig_vals <- eigen(A)$values

# det(A) = product of eigenvalues
det(A)
prod(eig_vals)  # Same!

# trace = sum of eigenvalues
sum(diag(A))
sum(eig_vals)  # Same!

# For numeric precision, use Re() on complex
A2 <- matrix(c(3, -1, 2, 5), nrow = 2, byrow = TRUE)
ev <- eigen(A2)$values
all.equal(det(A2), prod(Re(ev)), tolerance = 1e-10)
all.equal(sum(diag(A2)), sum(Re(ev)), tolerance = 1e-10)

Positive Definite Matrices

A symmetric matrix is positive definite (PD) if all eigenvalues are positive. Covariance matrices are positive semi-definite (eigenvalues ≥ 0). PD matrices are invertible and have Cholesky decompositions.

# Check positive definiteness
is_positive_definite <- function(A) {
  # Symmetric check
  if (!isTRUE(all.equal(A, t(A)))) return(FALSE)
  all(eigen(A)$values > 0)
}

# Positive definite covariance matrix
S <- matrix(c(4, 2,
              2, 3), nrow = 2, byrow = TRUE)
is_positive_definite(S)  # TRUE
eigen(S)$values  # Both positive

# Not PD (one negative eigenvalue)
Q <- matrix(c(1, 3,
              3, 2), nrow = 2, byrow = TRUE)
is_positive_definite(Q)  # FALSE
eigen(Q)$values  # One negative

Power Iteration for Dominant Eigenvalue

For large matrices, computing all eigenvalues is expensive. Power iteration finds the largest eigenvalue iteratively — the basis of Google's PageRank algorithm. Multiply by A repeatedly, then normalize.

# Power iteration: finds dominant eigenvalue
power_iteration <- function(A, tol = 1e-10, max_iter = 1000) {
  n <- nrow(A)
  v <- rnorm(n); v <- v / sqrt(sum(v^2))  # random unit vector
  lambda_old <- 0
  for (i in seq_len(max_iter)) {
    w <- A %*% v
    lambda <- max(abs(w))
    v <- w / lambda
    if (abs(lambda - lambda_old) < tol) break
    lambda_old <- lambda
  }
  list(value = lambda, vector = v)
}

A <- matrix(c(4, 1, 2, 3), nrow = 2, byrow = TRUE)
result <- power_iteration(A)
result$value  # Should be ~5 (dominant eigenvalue)
eigen(A)$values[1]  # Compare

Quick Check

Test your understanding of eigenvalues and eigenvectors in R.

Answer it in the interactive lesson

Recap: Eigenvalues and Eigenvectors

Key takeaways: eigen(A) returns $values and $vectors. Verify with A %*% v - lambda*v ≈ 0. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. Non-symmetric matrices may have complex eigenvalues. det(A) = product of eigenvalues; trace = sum. PCA decomposes the covariance matrix. Positive definite ↔ all eigenvalues > 0.

A <- matrix(c(6, 2, 2, 3), nrow = 2)
eig <- eigen(A)

# Key eigen operations:
eig$values    # eigenvalues
eig$vectors   # eigenvectors (columns)

# Verify Av = lambda*v
v1 <- eig$vectors[, 1]
max(abs(A %*% v1 - eig$values[1] * v1))  # ~0

# Properties
all.equal(det(A), prod(eig$values))       # TRUE
all.equal(sum(diag(A)), sum(eig$values))  # TRUE

# Reconstruct A = V Lambda V^{-1}
V <- eig$vectors
all.equal(A, Re(V %*% diag(eig$values) %*% solve(V)))

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