# 4: Eigenvalues and Eigenvectors

- Page ID
- 63397

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.

- 4.2: 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.

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)