The final test will be based on all material we covered during the semester.
However, proofs of theorems (or parts of theorems) will be restricted
to those covered after the midterm. Everything else (definitions and
problems) will cover all the material we studied. So, the midterm preparation
guide (minus proofs of theorems from the book) should be considered a part of
this guide. Below I will list only new material.

In addition to the theoretical material from the book, there will be
some other problems, not listed here.



Definitions and notions:
========================

Floating point axioms ( (13.5) and (13.7) )
Stability of an algorithm
Backward stability of an algorithm

Positive definite matrix
Eigenvalue, eigenvector, eigenspace, spectrum

Characteristic polynomial

Geometric multiplicity, algebraic multiplicity

Similarity transformation

Eigenvalue decomposition (diagonalization)

Defective eigenvalue, defective matrix

Diagonalizable matrix

Trace of a matrix

Schur factorization

Hessenberg matrix

Rayleigh quotient



Things to prove:
================

Given an m-by-n complex matrix A, and vector b,
prove that the value of the 2-norm of the residual r=Ax-b is minimized
when Ax=Pb, where P is the orthogonal projector onto the
range of A

Describe steps in the Householder triangularization algorithm


Thms 15.1, 16.2, 16.3.
Thm 17.1, case m=2

Give example of instability of LU decomposition
(and explain how this instability manifests itself).

Describe in details the LU decomposition
algorithm without pivoting. Prove the properties
of the elementary matrices that enable this simple algorithm
(the first two "strokes of luck").

Prove that is A is a Hermitian positive definite matrix,
then X*AX is also Hermitian positive definite.
(Here X is m by n matrix, m greater or equal than n, full rank.
X* is the adjoint of X.)

Prove that any principal submatrix of a positive definite matrix
is also positve definite.

Prove that every Hermitian positive definite matrix has a Cholesky
decomposition (one induction step is enough).

Prove thms:
24.3 24.4 24.5 24.6 24.9  27.1 28.1

Give an example of a defective matrix, with an explanation

Show that in the case of a real symmetric matrix
the eigenvectors are the stationary points of the Rayleigh quotient.
(compute the gradient).

Describe reduction to the Hessenberg form.
(Here and below "describe" means:
give the details of the algorithm and explain
how and why it works.)

Describe: power iteration, inverse iteration,
Rayleigh quotient iteration.