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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2019 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 2019 • Question 1
Let \(A\in\mathbb{C}^{m\times m}\) be hermitian: **a)** Prove that all eigenvalues of \(A\) are real. **b)** Prove that if \(x\) and \(y\) are eigenvectors corresponding to distinct eigenvalues, then \(x\) and \(y\) are orthogonal.

Raw LaTeX

Let \(A\in\C^{m\times m}\) be hermitian:

\partitem{a} Prove that all eigenvalues of \(A\) are real.

\partitem{b} Prove that if \(x\) and \(y\) are eigenvectors corresponding to
distinct eigenvalues, then \(x\) and \(y\) are orthogonal.

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Orthogonal and Unitary Matrices Projectors and Orthogonal Projections