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4: Eigenvalues and Eigenvectors

Last updated

May 28, 2023
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- 3.5.1: Exercises 3.5
- 4.1: Eigenvalues and Eigenvectors

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We have often explored new ideas in matrix algebra by making connections to our previous algebraic experience. Adding two numbers, $x + y$ , led us to adding vectors $\vec{x}+\vec{y}$ and adding matrices $A + B$ . We explored multiplication, which then led us to solving the matrix equation $A\vec{x}=\vec{b}$ , which was reminiscent of solving the algebra equation $ax = b$ .

This chapter is motivated by another analogy. Consider: when we multiply an unknown number $x$ by another number such as $5$ , what do we know about the result? Unless, $x = 0$ , we know that in some sense $5x$ will be “ $5$ times bigger than $x$ .” Applying this to vectors, we would readily agree that $5\vec{x}$ gives a vector that is “ $5$ times bigger than $\vec{x}$ .” Each entry in $\vec{x}$ is multiplied by $5$ .

Within the matrix algebra context, though, we have two types of multiplication: scalar and matrix multiplication. What happens to $\vec{x}$ when we multiply it by a matrix $A$ ? Our first response is likely along the lines of “You just get another vector. There is no definable relationship.” We might wonder if there is ever the case where a matrix – vector multiplication is very similar to a scalar – vector multiplication. That is, do we ever have the case where $A\vec{x}=a\vec{x}$ , where $a$ is some scalar? That is the motivating question of this chapter.

4.1: Eigenvalues and Eigenvectors
- 4.1.1: Exercises 4.1
4.2: Properties of Eigenvalues and Eigenvectors
In this section we’ll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalues; the goal here is not to memorize various facts about matrix algebra, but to again be amazed at the many connections between mathematical concepts.
- 4.2.1: Exercises 4.2

Thumbnail: A 2×2 real and symmetric matrix represent a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. (CC0; Jacopo Bertolotti via Wikipedia)

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