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Examiner: Prof. Rüde

Note: There were no exam protocols for this subject before my exam, so I created my own trainer for ANLA with 138 questions. It covers the lectures 1-15, as these are crucial for understanding the rest of the book. Take a look at it at http://kilians.net/trainers/anla/.

Atmosphere: Prof. Rüde was very kind, it was a relaxed atmosphere. He directly started with the first question. He did not focus on formulas, but on the general understanding of the concepts.

Questions:

Gerschgorin’s Theorem (from exercises)

• He gave me a real symmetric matrix and asked me, what the Gerschgorin’s Theorem says about the location of the eigenvalues.
• What can I say about them regarding the symmetry of the matrix?
• Draw a diagram with the regions of the eigenvalues.
• Can eigenvalues lie on the edge of the disks?
• What can you say about eigenvalues of singular matrices?

Norms

• What three conditions must hold for a norm?
• How is a matrix norm defined and what is its meaning?
• What fourth condition holds for matrix norms?

Least Squares

• What is a Least Squares problem?
• What are the three ways to solve it?
• How do the three ways compare regarding their complexity?
• How do the three ways compare regarding their numerical stability?

Eigenvalue Algorithms (the questions all referred to the lectures “Eigenvalue Problems” to “Reduction to Hessenberg Form” in the Book)

• What are the general steps to compute the eigenvalues of a matrix?
• Why can’t we directly compute the eigenvalues of a matrix that is not in Hessenberg form?
• Explain the simple version of the QR algorithm.
• How did we modify the QR Algorithm to get better results?

Preparation for the exam: I read the book (“Numerical Linear Algebra” by Trefethen and Bau) from lecture 1 to 31 a couple of times. I tried to fully understand the chapters 1-15 and 24-27. I understood the basic concepts of the lectures 16-23 and 28-31. Although Prof. Rüde covered 1-38 in the lectures, this was the optimal preparation.