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Chapter 7 will focus on matrices $A$ with the property that $A^{T}=A$ . Exercises 23 and 24 show that every eigenvalue of such a matrix is necessarily real.Let $A$ be an $n \times n$ real matrix with the property that $A^{T}=A$ . Show that if $A \mathbf{x}=\lambda \mathbf{x}$ for some nonzero vector $\mathbf{x}$ in $\mathbb{C}^{n},$ then, in fact, $\lambda$ is real and the real part of $\mathbf{x}$ is an eigenvector of $A$ . [Hint: Compute $\overline{\mathbf{x}}^{T} A \mathbf{x},$ and use Exercise $23 .$ Also, examine the real and imaginary parts of $A \mathbf{x} . ]$

$A x$ and $\lambda x$ are equal, their renl parts are also equal. $\rightarrow A u = \lambda u$ meaning that real part of $x$ is eigenvector of $A$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 5

Complex Eigenvalues

Vectors

Oregon State University

Harvey Mudd College

Idaho State University

Boston College

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In this example, we have a matrix a that contains entries that are real numbers and this square of size and buy in. We're going to assume that a transpose is equal to a and that a X equals Lambda X, where we have X is a non zero vector containing entries from the set of complex numbers. So with all those assumptions, what we're going to do is the following show that Lambda is in our meaning that it's a real number. And as per the hints provided, we already know that X transpose times a X is also a real number, no matter what the vector X happened to be, so it's use the information to start out will compute X transpose ex congregate transpose times a X and by substitution. We know X equals Lambda X. So let's make that substitution. Here we'll have ex con trick it transpose times Lambda X, and because land is a scaler, we can let that commute so that we have altogether Lambda Times the congregate of X transpose times X Then we know already that the contra kit of X transpose times a X is in our and we claim that this is following X congregate transpose Times X is also in our despite the fact that X comes from the set of and two pools of complex numbers. Let's give some justification to the last portion. Let's say let X be equal to Z one through ZN where these are now complex numbers. Then the congregate of X transpose Times X would be equal to the contra gave Z one times z one plus all the way through the conjure git of zn times e n and Z I conjure git Times e is in our four all I will for any complex number we have that its contra git times itself is purely a real number. So we know two things Now we know that ex contra get transpose times a X isn't is in our and we have no, by the last calculation, we see then that ex contra get transpose Times x is also in our So what does this mean? If we go back to this equation, this is purely a real number Now this is purely a real number And if Lambda was a complex number, then there'd be no way that this would be purely riel. So the conclusion that were forced to write is that this shows that Lambda is in our so all together. Knowing that a transpose equals a a X equals Lambda X is enough to show tell us that if X is not zero lambda must be a real number. We might wonder, where did this assumption come into play? Well, that was used to prove the hint here. Now the second thing we want to do here is to show that the real part of our Eigen Vector X is and I can vector of a to do this. Let's start out by letting X be equal to at Vector Z plus w I where z and W are in or in. So if we take a vector X from CNN, we can always decompose it in this way where the two Vectors NW came from our end. So now the exes decomposed. Let's write down something quick. So our equations were going to deal with are going to be eight times X which is now by substitution eight times z plus w i Well, this would be a Times Z, but recall that Z is also the real part of X, then we'll add to that eight times. W i. But W is the imaginary part of X times. I Let's look at this equation from a different point of view. We can also calculate a X equals lamb Dax provided by this equation here. Then, if we substitute at this stage, will have Lambda Times, z plus W Y in place of X. And now if we distribute, it follows that will have Lambda Times Z. But recalled Z is the real part of X so will express it in this way, plus Lambda Times W where w is the imaginary part of X times I now with the equation we've written, we have equality that goes here. And the only way that these two expressions can be equal is if they're riel and imaginary parts are equal. And this forces this quantity to be equal to each other. And so we have now this conclusion this implies the A times The real part of X, which is a non zero vector, is equal to Lambda Times, the real part of X, and this shows that the real part of X is an Eigen vector of a as required

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