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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2022 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Fall 2022 • Question 3

(Hadamard’s Inequality). Prove algebraically: Let \(A \in \mathbb {C}^{m \times m}\) and let \(\mathbf {a}_j\) denote the \(j^{\text {th}}\) column of \(A\). Then, show that, \[ |\det (A)| \le \prod _{j=1}^m \|\mathbf {a}_j\|_2. \]

Raw LaTeX

(Hadamard's Inequality).
Prove algebraically:
Let \(A \in \mathbb{C}^{m \times m}\) and let \(\mathbf{a}_j\) denote the \(j^{\text{th}}\) column of \(A\). Then, show that,
\[
|\det(A)| \le \prod_{j=1}^m \|\mathbf{a}_j\|_2.
\]

Topic Tags

Determinants and Hadamard's Inequality Linear Programming and Optimization