I'm completely lost on how to finish this problem. The problem from the book is
Find a matrix $P$ that orthogonally diagonalizes $A$ and determine $P^{-1}AP$
and my matrix is the $2 \times 2$ matrix $$\begin{pmatrix} 6 & 2\sqrt{3} \\ 2\sqrt{3} & 7 \end{pmatrix}$$
So far I've found the characteristic equation to be $x^2-13x+30 = 0$ and have factored it to be $(x-10)(x-3)=0$ and then found the eigenvalues of $\lambda=3, 10$. My diagonal matrix then is $$D=\begin{pmatrix} 3 & 0 \\ 0 & 10 \end{pmatrix}.$$
From this I'm supposed to find $P$ and $P^{-1}$, my values for $P$ have been wrong this whole time and I end up with
$$P=\begin{pmatrix}
2\frac{\sqrt{3}}{3} & 1 \\
\frac{\sqrt{3}}{4} & 1
\end{pmatrix}$$
The book says
$$P=\begin{pmatrix}
-\frac{2}{\sqrt{7}} & \frac{\sqrt{3}}{\sqrt{7}} \\
\frac{\sqrt{3}}{\sqrt{7}} & \frac{2}{\sqrt{7}}
\end{pmatrix}.$$
I tried using the formula $\frac{\vec{v_1}}{||\vec{v_1}||}$ that looked relevant to this from the book, but I didn't end up with what the book had.
I really appreciate any help!
-Frank
Best Answer
Hint: Find the eigenvectors that correspond to $\lambda=3,10$ and then normalize them by dividing by their norm - this is necessary to produce an orthogonal matrix, which has orthonormal columns (column vectors that form an orthogonal set and are unit vectors). Refer to my answer on the following post for the form of $P$: Some techniques to test for diagnalizibility for some random Matrix, lets say A?