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In Exercises $7-12$ , use Example 6 to list the eigenvalues of $A$ . In each case, the transformation $\mathbf{x} \mapsto A \mathbf{x}$ is the composition of a rotation and a scaling. Give the angle $\varphi$ of the rotation, where $-\pi<\varphi \leq \pi,$ and give the scale factor $r .$$$\left[\begin{array}{cc}{\sqrt{3}} & {3} \\ {-3} & {\sqrt{3}}\end{array}\right]$$

$A = \left[ \begin{array} { c c } { \sqrt { 3 } } & { 3 } \\ { - 3 } & { \sqrt { 3 } } \end{array} \right]$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 5

Complex Eigenvalues

Vectors

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Okay, So for problem eight were given the Matrix Square with three three and negative three's good three. So, by our result, in the example six. Now he tom along more textbook. Um, we know the Lando. Here we take. So first here we take a to B square three B to B. Negative three. So our landa will be, but with me minus or a plus, um three. Okay, so for the scaling and the rotation, So first we need to find out are which is the land of Lunda. So that's three plus nine. Take the square root. Just a square root. Tough 12 Two times square three. Okay. And so See, we take out. Squared off three. We will have one have square with three over two. Negative square was three over two and one have now the and go. Just take negative three by over three. So no sign off. Connective high over three will be 1/2 and minus. Sign active by over three. Wilkie Positive square root. Three over two. Sign of collective fly over three. But just to be inactive were three over two and again Course I am negative. Piled on three will be 1/2. So here we take you to the connective. High over three

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