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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2018 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Spring 2018 • Question 2

Hadamard’s inequality

Let \(A\in \mathbb {C}^{N\times N}\) be comprised of columns \(a_i\in \mathbb {C}^N\), \(i=1,\ldots ,N\). Prove that \[ |\det A|\le \prod _{i=1}^{N}\|a_i\|_2. \] You may use, for example, a QR decomposition/Gram-Schmidt argument.

Raw LaTeX

{Hadamard's inequality}

Let \(A\in\C^{N\times N}\) be comprised of columns \(a_i\in\C^N\),
\(i=1,\ldots,N\). Prove that
\[
  |\det A|\le \prod_{i=1}^{N}\|a_i\|_2.
\]
You may use, for example, a QR decomposition/Gram-Schmidt argument.

Topic Tags

Determinants and Hadamard's Inequality Linear Programming and Optimization QR Factorization Singular Value Decomposition