Noori Selina Discussion #3

Chapter E

C23

C23 Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for A = 1 1 1 1

First we define the matrix

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

Eigenvalues and Eigenvectors are then calculated

eigen_A <- eigen(A)
print(eigen_A)

## eigen() decomposition
## $values
## [1] 2 0
## 
## $vectors
##           [,1]       [,2]
## [1,] 0.7071068 -0.7071068
## [2,] 0.7071068  0.7071068

Extracting eigenvalues and eigenvectors

lambda <- eigen_A$values
vectors <- eigen_A$vectors

print("Eigenvalues:")

## [1] "Eigenvalues:"

print(lambda)

## [1] 2 0

print("Eigenvectors:")

## [1] "Eigenvectors:"

print(vectors)

##           [,1]       [,2]
## [1,] 0.7071068 -0.7071068
## [2,] 0.7071068  0.7071068

Creating an empty list to store the eigenvectors that corresponds to each eigenvalue

eigenspaces <- list()

Computing the eigenspaces and geometric multiplicities by iterating through each eigenvalue, extracting the corresponding eigenvectors, and determining the number of similar eigenvalues, hence representing geometric multiplicities.

for (i in 1:length(lambda)) {
  eigenspaces[[i]] <- vectors[, i]
  geometric_multiplicity <- sum(abs(lambda[i] - lambda) < 1e-10)
  print(paste("Eigenvector", i, ":"))
  print(eigenspaces[[i]])
  print(paste("Geometric multiplicity for eigenvalue", lambda[i], ":", geometric_multiplicity))
  print(paste("Eigenspace for eigenvalue", lambda[i], ":"))
  print(eigenspaces[[i]])
}

## [1] "Eigenvector 1 :"
## [1] 0.7071068 0.7071068
## [1] "Geometric multiplicity for eigenvalue 2 : 1"
## [1] "Eigenspace for eigenvalue 2 :"
## [1] 0.7071068 0.7071068
## [1] "Eigenvector 2 :"
## [1] -0.7071068  0.7071068
## [1] "Geometric multiplicity for eigenvalue 0 : 1"
## [1] "Eigenspace for eigenvalue 0 :"
## [1] -0.7071068  0.7071068