6610-2021-08

(Matrix properties) For each of the following statements, either prove that it is true, or show it’s false with a counterexample.

a. For \(A \in \mathbb {C}^{m \times n}\) any matrix, and \(U \in \mathbb {C}^{m \times m}\) any unitary matrix, then \(\|UA\|_2 = \|A\|_2\). b. For \(A \in \mathbb {C}^{n \times n}\) any matrix, if \(\rho (A) = 0\), then \(A = 0\). (Recall that \(\rho (A)\) is the spectral radius of \(A\), i.e., the maximum of the moduli of the eigenvalues of \(A\).)

c. For \(A \in \mathbb {C}^{m \times n}\) any matrix with \(m \ge n\), then the squared singular values of \(A\) equal the eigenvalues of \(A^*A\).

(Moore-Penrose pseudo-inverse)

Let \(A \in \mathbb {C}^{m \times n}\) have rank \(r\) and reduced SVD \[ A = \tilde {U}\tilde {\Sigma }\tilde {V}^* . \]

The Moore-Penrose pseudoinverse of \(A\) is defined as

a. In the induced matrix \(\ell ^2\) norm \(\|\cdot \|\), is the operation \(A \mapsto A^+\) well-conditioned? That is, given an arbitrary but fixed \(A\) and a perturbation matrix \(B\), is \(\|(A + B)^+ - A^+\|/\|A^+\|\) controllable by \(\|B\|/\|A\|\)? Prove it, or give a counterexample.

b. If \(m \ge n\) and \(A\) has full rank \(n\) and \(b \in \mathbb {C}^m\) is a given vector, prove that \(x = A^+b\) is identical to the least-squares solution \(z\) of the over-determined linear system \(Az = b\).

(Finite difference formulas) Given \(h > 0\), compute weights \(\{w_j\}_{j=0}^2\) for the following one-sided finite difference formula for the first derivative \(f'(x)\): \[ f'(x) \approx \sum _{j=0}^2 w_j f(x+jh) = w_0 f(x) + w_1 f(x+h) + w_2 f(x+2h). \]

What is the order of accuracy of your formula?

(Quadrature rules) Compute the weights \(\{A_0, A_1, B_0, B_1\}\) for a 4-point quadrature rule of the form \[ \int _0^1 f(x)\,dx \approx A_0 f(0) + A_1 f(1) + B_0 f'(0) + B_1 f'(1) \]

that is exact when \(f\) is any polynomial of degree 3 or less.