403/503 – HW7

April 16, 2015

This exercise is due Tuesday, April 21, at the beginning of lecture.

Find the Singular Value Decomposition of

\displaystyle M=\left(\begin{array}{ccccc}1&0&0&0&2\\ 0&0&3&0&0\\ 0&0&0&0&0\\ 0&4&0&0&0\end{array}\right).

(I am not so interested in the specific answer, which can be found online, but rather in the process describing how one arrives to this answer.)

403/503 – Another extra credit problem

April 9, 2015

This optional homework is due Thursday, April 16, at the beginning of lecture. We want to verify the details of Jacobi method for diagonalizing symmetric matrices.

Recall that the Givens rotation R(i,j,\theta) has the effect of rotating a vector by \theta radians in the i,j-plane.

Suppose A=(a_{k,l})_{k,l=1}^n is an n\times n real symmetric matrix, and that i\ne j. Let B=(b_{k,l}) be the matrix G^T A G where G=R(i,j,\theta).

  1. Show that if \theta is chosen so that \displaystyle\cos\theta=\left(\frac12+\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2} and \displaystyle\sin\theta=\left(\frac12-\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2}, where \displaystyle \beta=\frac{a_{ii}-a_{jj}}{2a_{ij}}, then b_{i,j}=b_{j,i}=0.
  2. Show that \displaystyle\sum_{k=1}^nb_{k,k}^2=2a_{i,j}^2+\sum_{k=1}^na_{k,k}^2.
  3. Show that \displaystyle\sum_{k,l}b_{k,l}^2=\sum_{k,l}a_{k,l}^2.

403/503 – Extra credit problem

April 8, 2015

This problem is due April 30 at the beginning of lecture.

Write a program that receives as input a real symmetric matrix A and some tolerance bound \epsilon, and performs the basic QR method to A generating (and printing) a sequence of matrices A_0=A,A_1,A_2,\dots until a stage n is reached where the entries below the diagonal  of A_n are all in absolute value below \epsilon. Once this happens, the program returns the diagonal entries of A_n as approximations to the eigenvalues of A. (Check on a couple of examples that these are indeed decent approximations, at least for A of small size and reasonably small values of \epsilon.)

Most Computer Algebra Systems already have implemented algorithms to find the QR decomposition of a matrix. Instead of using these pre-programmed algorithms, write your own.

(Turn in the code, plus the couple of examples. Comment your code, so it can be easily understood what you are doing along the way. I’m reasonably familiar with Maple, Mathlab, Sage, and most flavors of C. If you are going to use a different language, please let me know as soon as you can, to see whether it is something I’ll be able to verify or if a different language will be needed instead. Ideally, the user can choose the dimension of the input matrix.)

311, 403/503 – Update

March 1, 2015

Our daughter decided to rush things up a bit, and is being born right now. You probably will have someone else covering class for a couple of weeks, sorry for the inherent inconvenience. I’ll be updating as I know more.

[Edit (11:00am): Isabel. Office hours are cancelled March 3 and 10.]

403/503 – HW6

February 24, 2015

This exercise is due Tuesday, March 3, at the beginning of lecture.

Recall that the nth Jordan block for \lambda, J(\lambda,n), is the n\times n matrix whose entries along the main diagonal are \lambda, along the diagonal immediately below the main one are 1, and all other entries are 0. For example, J(5,4) is the matrix

\displaystyle \left(\begin{array}{cccc}5&0&0&0\\ 1&5&0&0\\ 0&1&5&0\\ 0&0&1&5\end{array}\right).

Find a general formula for the powers of Jordan blocks, i.e., compute J(\lambda,n)^k.

403/503 – Extra credit homework

February 5, 2015

This set is optional, and due February 12 at the beginning of lecture.

From Chapter 3 of Axler’s book, solve exercises 1,2,3,9,10,16, and 25.

(The numbering is as in the second edition. If you own the third edition, let me know and I’ll check in case the statements have changed.)

403/503 – HW5

January 29, 2015

Show that affine spaces are closed under affine combinations, that is: If C is an affine space, n is any positive integer, c_1,\dots,c_n are any vectors in C, and r_1,\dots,r_n are any reals such that


then r_1c_1+\dots+r_nc_n\in C.

(Due February 2 at the beginning of lecture.)

403/503 – HW4

January 27, 2015

Let b\in\mathbb R^m and let A be an m\times n matrix with real entries. Set C=\{x\in\mathbb R^n\mid Ax=b\}, and suppose that C\ne\emptyset. Show that C is an affine space.

(Due January 29 at the beginning of lecture.)

403/503 – HW3

January 23, 2015

Recall that a vector space V is said to be of dimension n, \dim V=n, iff there is an independent subset of V of size n, and no independent subset has size n+1.

A basis of a vector space V is any independent set whose span is V.

Suppose that V is a vector space of dimension n. In lecture we showed that a subset of V of size n is independent iff it spans V. Show the following:

  1. V has a basis, and all bases have size n.
  2. If A\subseteq V is independent, then there is a basis B of V with A\subseteq B.

(Due January 27 at the beginning of lecture.)

403/503 – HW2

January 20, 2015

Let V be a vector space. In lecture we verified that the following two statements about a set A\subset V are equivalent:

  • For any v_1,\dots,v_n\in A and any scalars r_1,\dots,r_n and s_1,\dots,s_n, if \displaystyle \sum_{i=1}^n r_i v_i=\sum_{i=1}^n s_i v_i, then r_i=s_i for all i.
  • For any v_1,\dots,v_n\in A and any scalars r_1,\dots,r_n, if \displaystyle \sum_{i=1}^n r_i v_i=0, then r_i=0 for all i.

Recall that the set A is independent iff no element of A is in the span of the other elements, that is, for any a\in A, we have that a\notin\mathrm{sp}(A\setminus\{a\}).

  1. Show that A is independent iff the two (equivalent) statements above hold.

(Due January 22 at the beginning of lecture.)