Discover Linear Algebra: A first course in linear algebra

We have seen that when considering a specific matrix $A\text{,}$ looking for patterns in the process of computing matrix-times-column-vector helps us to understand the matrix. In turn, this helps us understand all of the various systems $A\mathbf{x}=\mathbf{b}$ with common coefficient matrix $A\text{,}$ since obviously the left-hand side of the matrix version of the system has matrix-times-column-vector form.

When we compute $A{\mathbf{e}}_j$ for a standard basis vector ${\mathbf{e}}_j\text{,}$ the result is the $j^{\mathrm{th}}$ column of $A\text{.}$ So if we computed each of $A{\mathbf{e}}_1,A{\mathbf{e}}_2,\dots ,A{\mathbf{e}}_n\text{,}$ we would have all of the columns of $A$ as the results, which contain all of the data contained in $A\text{.}$ These computations certainly let us know the matrix $A\text{,}$ but they don’t necessarily help us understand what $A$ is really like as a matrix. In short, the standard basis for ${\mathbb{R}}^n$ is a great basis for understanding the vector space ${\mathbb{R}}^n\text{,}$ but it is not so great for helping understand matrix products $A\mathbf{x}$ for a particular matrix $A\text{.}$

A complete study of how the concepts of eigenvalues and eigenvectors unlock all the mysteries of a matrix is too involved to carry out in full at this point, but we will get a glimpse of how it all works for a certain kind of square matrix in the next chapter. For the remainder of this chapter, we will be more concerned with how to calculate eigenvalues and eigenvectors.