STAT151A Code homework 2: Due February 9th

In this homework, you’ll implement some linear algebra ideas in R.

For each problem, provide a solution in the form of a function, and then test it using following provided functions:

# Check whether matrix `x` == `y` for all entries,
# up to the tolerance `tol`.  If not, an error is raised.
AssertMatricesEqual <- function(x, y, tol=1e-9) {
  if (!(all(dim(x) == dim(y)))) {
    stop("The dimensions do not match.")
  }
  err <- max(abs(x - y))
  if (err > tol) {
    stop(sprintf("The error %f is greater than the tolerance.", err))
  }
}


# Generate an `nrow` x `ncol` matrix with random standard normal entries.
GenerateMatrix <- function(nrow, ncol) {
  return(rnorm(nrow * ncol) %>% matrix(nrow, ncol))
}

For example,

# Should fail --- the matrices have the wrong dimension.
AssertMatricesEqual(GenerateMatrix(3, 3), GenerateMatrix(4, 3))

# Should fail --- the matrices are not equal.
A <- GenerateMatrix(3, 3)
AssertMatricesEqual(GenerateMatrix(3, 3), A + 5)

# Should pass
AssertMatricesEqual(A, A)

1. (Example) Starting from GenerateMatrix(), write a function that generates a random symmetric matrix.

Solution:

GenerateSymmetricMatrix <- function(dim) {
    a_mat <- GenerateMatrix(dim, dim)
    return(0.5 * (a_mat + t(a_mat)))
}

a_sym <- GenerateSymmetricMatrix(4)
AssertMatricesEqual(a_sim, t(a_sim))

1. (Example) Starting from AssertMatricesEqual(), write a function that checks whether a matrix is symmetric.

Solution:

CheckSymmetricMatrix <- function(a) {
    AssertMatricesEqual(a, t(a))
}

1. Starting from GenerateMatrix(), write a function to generate a random symmetric positive semi-definite matrix of a given size.

2. Starting from GenerateMatrix(), write a function to generate a random symmetric positive definite matrix of a given size whose smallest eigenvalue is greater than one. (Hint: you can add something to your previous solution.)

3. Write a function that takes in a symmetric PSD matrix and returns an inverse computed from the eigendecomposition. You may use eigen().

4. Write a function that takes in a symmetric PSD matrix and returns a square root computed from the eigen-decomposition. You may use eigen().

5. Write a function that takes in a symmetric PSD matrix and returns a (possibly non-symmetric) square root computed from the cholesky decomposition. You may use chol().

6. Write a function that takes in a potentially non-square matrix and returns a projection matrix onto the column span. You may assume the matrix is full-rank.

7. Write a function that takes in a potentially non-square matrix and returns a projection matrix onto the row span. You may assume the matrix is full-rank.

8. Write a function that takes in a potentially non-square matrix and returns a projection matrix onto the orthogonal complement of the column span. You may assume the matrix is full-rank.

9. Write a function that takes in a potentially non-square matrix and returns an orthonormal basis for its column span. You may assume the matrix is full-rank.

10. Write a function that takes in a positive definite covariance matrix and returns a draw from the multivariate normal distribution with mean zero and the given covariance matrix. You may use only rnorm(n, mean=0, sd=1) to generate random variables. Verify that you have succeeded using Monte Carlo draws and AssertMatricesEqual() with an appropriate tolerance.