Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2021 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2021 • Question 1

(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 \(A^*A\).

(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 \(A^*A\).

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra Singular Value Decomposition