My Math Forum Eigenvalues problem

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 December 22nd, 2009, 01:56 AM #1 Newbie   Joined: Dec 2009 Posts: 21 Thanks: 0 Eigenvalues problem Plz help me to find the answer for following question; A matrix $A= \begin{bmatrix} 16 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{bmatrix}$ Find (a) Find Eigen valves A (b) Egen vectors A (c) obtain a matrix P such that $P^{-1} AP$ is a diagonal matrix (d) If $S=P^{-1} AP$, state the special property of each of $S$ and $S^{-1}$ (e) Using the above result reduce the quadratic from Q(x) to form Q(y) where $Q(x)=16x_1^2 + \frac{29}{5} x_2^2 + \frac{36}{5} x_3^2 + \frac{24}{5} x_2 x_3$ $Q(y)=16y_1^2 + +9y_2^2 + 4y_3^2$ (f) obtain the relationship between $x^t= (x_1 ,x_2 ,x_3)^t$ and $y^t (y_1 ,y_2 ,y_3)^t$ Now I solved part (a) as follows $A- \lambda I=\begin{bmatrix} 16- \lambda & 0 & 0 \\ 0 & \frac{29}{5} - \lambda & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} - \lambda \end{bmatrix}$ the result gain $( \lambda -16)( \lambda -4)( \lambda -9)$ and part (b) solved as follows $\begin{pmatrix} 16 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = 16$ then if $\lambda=4$ also I did for $\lambda=9$ now I need help to find rest of parts in this question, also I need to know the above calculations are in correct or not!! if some are in incorrect plz. let me know the place and how I correct it

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