# 4: Eigenvalues and Eigenvectors


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.

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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