Study guide for Examlet 4
Here is a non-exhaustive list of questions you should be able to answer as you prepare for the examlet.
Past chapters
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(Recall from the course policies that our examlets are cumulative.)
Graphs and Sparsity
- What is an adjacency matrix?
- How does a graph get represented as a matrix?
- What does matrix-vector multiplication with an adjacency matrix mean?
- What is a sparse matrix?
- How does CSR format for the representation of sparse matrices work?
- What would matrix-vector multiplication with CSR matrices look like?
- What is the computational cost of working with sparse matrices?
Norms and Conditionining
- What criteria does a vector norm have to satisfy?
- What is the triangle inequality?
- What are the $p$-norms?
- What is the "unit ball" of a norm?
- What is a matrix norm? submultiplicativity?
- How can the matrix norm of a diagonal matrix be computed?
- What is special about matrix norms of orthogonal matrices?
- What is the condition number of solving a linear system? matrix-vector multiplication?
- What is the condition number of a matrix?
- How can the condition number of a diagonal matrix be calculated?
- How does a condition number affect the number of accurate digits in a result?
LU
- What do forward/backward substitution accomplish? How? At what computational cost?
- What are elimination matrices?
- How can they be inverted? multiplied by one another?
- What happens when you multiply an elimination matrix by another matrix? a vector?
- What LU factorization? How does it work? What is its computational cost?