Consider the 2 !! 4 matrix A = [0 0 3 0; 1 0 0 1] and let r = rank(A). We will work our way toward the Singular Value Decomposition (SVD) of A and to the pseudoinverse of A.
a.) Briefly explain why we know, without having to do any computations, that A^T A has at least two eigenvalues equal to 0
b.) Calculate A^T A. [Hint: Only five entries are nonzero]
c.) Show that det(A^T A – λ I) = λ^4 – 11λ^3 + 18λ^2.
d.) Compute the eigenvalues of A^T A. Label and order them as λ1 ≥ λ2 ≥ λ3 ≥ λ4.
e.) Compute the singular values of A. Label and order them as σ1 ≥ σ2 ≥ σ3 ≥ σ4.
f.) What is the matrix Σ1 for the reduced SVD of A? What is the matrix Σ for the SVD of A?
g.) For all of the nonzero eigenvalues of A^T A, find corresponding unit eigenvectors.
h.) Compute rref(A^T A) and find the "special solutions".
i.) Apply the Gram-Schmidt Process to the vectors you found in part (h).
j.) What is the matrix V1 for the reduced SVD of A? What is the matrix V for the SVD of A?
k.) For i = 1, . . . , r, compute the vectors ui.
l.) For i = r + 1, . . . , m, compute the vectors ui. If there are no such vectors, briefly explain why.
m.) What is the matrix U1 for the reduced SVD of A? What is the matrix U for the SVD of A?
n.) What is the matrix Σ+ for computing the pseudoinverse of A?
o.) Which matrix multiplication would you perform to compute the pseudoinverse of A? (Do not perform the multiplication)