This set is due Thursday, May 8, at the beginning of lecture. (There will be another homework set, due the scheduled day of the final exam, Thursday May 15, at 11am, so I recommend you try to complete this set earlier than the scheduled deadline.)

You can work on your own, or in groups of up to three members. In case you cannot find anybody to work with, and do not know how to program, let me know as soon as possible, and we will find an alternative. As usual, you can still collaborate with others not in your group, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc.

1. Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)

2. Write a program that, given a square matrix (diagonalizable and) with real entries, computes approximations to its eigenvalues using the -algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the (tolerance) error within which the approximations will be found. Apply your method to a symmetric matrix, and check the number of iterations the process requires, as a function of the tolerance error.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. To help verify that your algorithm is proceeding correctly, at each step of the iteration have your program indicate clearly what the matrices and are, and what the new (output) matrix is.

Please make the algorithm as explicit as possible. Meaning: Do not use shortcuts already built into the software; most CASs already have functions that perform the Gram-Schmidt process to a given set of vectors, or functions that give the decomposition of a matrix. Instead, I want you to program these subroutines as well.

The programming language you use is up to you. Maple, Mathlab, Sage are standard choices, but if you prefer a different language, it should be fine. Let me know, just in case.

3. Do the same, but now for Francis’s algorithm. Apply it to the same matrix. (Here there are more matrices and some vectors the algorithm may want to display along the way. For instance, whenever a matrix is put into upper Heissenberg form, indicate what the reflectors used along the way are.)

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One Response to 403 – HW 3 – Computing eigenvalues

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]

[…] 15. Reflectors. Francis’s algorithm (conclusion). Homework 3, due May […]