C&O 367, Winter 2001
Assignment 2

Due on Thursday, Jan. 25, (at start of class)
Instructor H. Wolkowicz

1. (10 marks) (Text: Problem 31, page 36)
   1. Let $A$ be an $n \times n$ symmetric matrix. Diagonalize $A$ to show that (the Raleigh quotient)
      
      $$\frac{x^tAx}{\vert\vert x\vert\vert^2}$$
      
      
      is greater than or equal to the smallest eigenvalue of $A$ for all $x \neq 0$ in $\Re^n$.
   2. Show that the quadratic form $Q_A(x) = x^tAx$ is coercive if and only if $A$ is positive definite.
   3. Conclude from 1b that if
      
      $$f(x) = a + b^tx + \frac 12 x^t A x$$
      
      
      is any quadratic function where $a \in \Re,~ b \in \Re^n$ and $A$ is an $n \times n$ symmetric matrix, then $f(x)$ is coercive if and only if $A$ is positive definite.

2. (5 marks) Suppose that
   
   $$f(x) = b^tx + \frac 12 x^t A x$$
   
   
   where $b \in \Re^n$ and $A$ is an $n \times n$ symmetric matrix. Show that $f(x)$ is bounded below on $\Re^n$ if and only if the minimum of f on $\Re^n$ is attained (i.e. there exists $\bar{x}$ such that $f(\bar{x})=\min\limits_{x\in\Re^n} f(x)$).
3. (5 marks) (Text: Problems 1a and 1d, page 77)
4. (5 marks) (Text: Problems 2b and 2c, page 77)
5. (5 marks) (Text: Problems 9, page 78)