Exercises:

1. Prove the following:
   a) $\vert\vert x\vert\vert _{\infty} \leq \vert\vert x\vert\vert _{1}\leq n\vert\vert x\vert\vert _{\infty}$
   b) $\vert\vert x\vert\vert _{\infty} \leq \vert\vert x\vert\vert _{2}\leq \sqrt{n}\vert\vert x\vert\vert _{\infty}$
2. Let A be a real non singular matrix of order n, and let $\vert\vert.\vert\vert _{\upsilon}$ denote a vector norm on $R^n$. Define
   
   $$\vert\vert x\vert\vert _{*}=\vert\vert Ax\vert\vert _{\upsilon} \qquad\qquad x\, \epsilon \, R^n$$
   
   
   Show that $\vert\vert.\vert\vert _{*}$ is a vector norm on $R^n$.
3. Let          $A(\alpha)=\left[% \begin{array}{cc} 0.1\alpha & 0.1\alpha \\ 1.0 & 1.5 \\ \end{array}% \right]$
   Determine $\alpha$ such that Cond$(A(\alpha))$ is minimised. Use the maximum norm.
4. Solve the following systems
   
   1. $4x_1+x_2+x_3=4$
      $x_1+4x_2-2x_3=4$
      $3x_1+2x_2-4x_3=6$
   2. $x_1+x_2-x_3=2$
      $2x_1+3x_2+5x_3=-3$
      $3x_1+2x_2-3x_3=6$
   by Gauss elimination i) without pivoting and ii) with partial pivoting.
5. Find the Inverse of the matrix
   
   
   $$\left[% \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 3 & -1 \\ 2 & -1 & 3 \\ \end{array}% \right]$$
   
   
   by Gauss-Jordan method.
6. The following system of equation is given
   
   $$4x+y+2z=4$$
   
   
   
   
   $$3x+5y+z=7$$
   
   
   
   
   $$x+y+3z=3$$
   
   
   1. Set up the Jacobi and Gauss-Seidel iterative schemes for the solution and iterate three times starting with the initial vector $x^{(0)}=0$. Compare with the exact solution.
7. The system of equations $Ax=b$ is to be solved iteratively by
   
   $$x_{n+1}=Mx_n+b$$
   
   
   Suppose $A=\left[% \begin{array}{cc} 1 & k \\ 2k & 1 \\ \end{array}% \right]$, $k \neq \frac{\sqrt{2}}{2}$, k is real.
   1. Find a necessary and sufficient condition on k for convergence of the Jacobi method.
8. Consider the tridiagonal matrix given by
   
   $$A=\left[% \begin{array}{ccccc} 4 & 1 & & & \\ 1 & 4 & 1... ... \\ & & 1 & 4 & 1 \\ & & & & 4 \\ \end{array}% \right]$$
   
   
   Show that A is nonsingular. Find a bound for $\vert\vert A^{-1}\vert\vert _{\infty}$ and $\vert\vert A^{-1}\vert\vert _2$.
9. Show using elimination that the following system does not have a solution:
   
   $$x_1+2x_2+x_3=3$$
   
   
   
   
   $$2x_1+3x_2+x_3=5$$
   
   
   
   
   $$3x_1+5x_2+2x_3=1$$
   
   

10. For the system $Ax=b$ with
    
    $$A=\left[% \begin{array}{ccc} 5 & 6 & 7 \\ 6 & 5 & 6 \\ 7 & 5 & 6 \\ \end{array}% \right]$$
    
    
    1. Find det.(A) and $\vert\vert A\vert\vert$. Is A ill-Conditioned?
    2. For $b=(18,17,18)$, the solution is $x=(1,1,1)$. Solve the system for the vectors $b=(17,17,17)$ and $b=(18.2,17.2,18.2)$ and observe how the solution change.