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Describe how you might try to build a solution of a difference equation $\mathbf{x}_{k+1}=A \mathbf{x}_{k}(k=0,1,2, \ldots)$ if you were given the initial $\mathbf{x}_{0}$ and this vector did not happen to be an eigenvector of $A \cdot\left[\text { Hint: How might you relate } \mathbf{x}_{0} \text { to eigenvectors of } A ?\right]$

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Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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Hello. I want to see in question. We have toe. We have to find how to try to write the equation. Difference equation. Right. So we could try to write X not as linear combination off Eigen vectors. We want we toe upto VP. If Lambda one if lamb down upto lambda pr the corresponding Aiken various. And if x not is a questo Si won, we won plus c two e two plus upto cp weepy right. Then we can define XK is a questo. C one lambda, one k, we went plus Saito Lambda. Okay, we to let's have toe cp Lambda Big, weepy like this case for judo 12 Okay, right. So we can multiply. He both sides. You can write it as see even Lambda One. Okay, everyone, let's see. Two lambda. Okay. Every two plus up to say p lend a big A V p right. We can like this. See, one lambda gave from the definition we can like toe. From the definition we can write with us on Stephen Lambda K plus one. We even let's say, do Lambda Tau K plus one way to similarly this'd. If we see, we can write it. Us XK plus one. Okay, So, like like this, we can try to write the difference equation. Okay. Thank you.

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