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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 .$$$\left[\begin{array}{rr}{1} & {.1} \\ {-1} & {.1}\end{array}\right]$$

$A = \left[ \begin{array} { c c } { .1 } & { .1 } \\ { - .1 } & { .1 } \end{array} \right]$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 5

Complex Eigenvalues

Vectors

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in this example. We have a long theorem here that's provided, and we're going to walk through it by illustrating its use with the Given Matrix. So, to that end, let's start off by letting A Matrix A, which is of size to buy two, to be defined by 0.1 and negative 0.1 In Column one, then 10.1 and 2.1 in Column two. So now with our Matrix say, Let's compare it with the Matrix C provided in the serum. The key feature with this matrix is that the main diagonal is equal, and that's what we have going on here with this matrix A and likewise, the off diagonal must be opposites of each other, and that's also what's provided here. So since the theorem applies to this matrix, say, let's list what the A and B should be. In this case will have that a is 0.1 coming from this entry and column one, and likewise be is negative 0.1 coming from the century here. Then, with that information, we're ready to state the Eigen values. The first portion of the theorem we have that the first Eigen value Lambda one is a plus B I. In our case, that's 10.1 minus 0.1 I, then lambda to is It's conjure git, which is 0.1 plus 0.1 I. So this is the first portion off this theorem for the second portion. It provides us with the tools of using this matrix factory ization for our particular matrix. A. But here are mingle is to just determine what the modules are should be as well as the argument fee. So we're solving for two things. Next, let's start by solving for our so in this case, are is the module ist of either Eigen value. The minus and plus in the middle don't give us a different value, and so this is going to be by definition of module ISS, the square root of 0.1 squared plus negative 0.1 squared, so this becomes 0.1 plus 0.1 which is the square root of 0.2 and it's always difficult to determine how much we want to reduce here. But let's reduce giving an exact value. So we have 0.2 or, in other words, that is to one hundreds, and we can say that that squirt of two in the numerator divide by 10. So this is our module. ISS are next. We're going to determine what fee is, but we have to use the full equation to start out. They're right. That tension tiffy is equal to be divide by a But we get that information here, so take negative 0.1 divided by 0.1. Now we get a negative one quantity altogether. Now, when we're looking for fee, we're just looking at what is the rotation for either Aiken Value? We're focusing on the first case, Lambda One. And for this first Eigen value noticed that ends up in the fourth quadrant. That that implies that he must be equal to negative pira for since tension. A native PIRA four does indeed give us negative one. And that also sets us in the right quadrant. So to summarize for this matrix say that's been provided thes are it's Eigen values. And if we're going to use that matrix Agon factory ization indicated above, this would be the module iss and this would be the argument

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