DLA Motivation

Section 24.3 Motivation

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

$A\uvec{x}=\uvec{b}$ 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

$A\uvec{e}_j$ for a standard basis vector

$\uvec{e}_j\text{,}$ the result is the

$\nth[j]$ column of

$A\text{.}$ So if we computed each of

$A\uvec{e}_1,A\uvec{e}_2,\dotsc,A\uvec{e}_n\text{,}$ 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

$\R^n$ is a great basis for understanding the vector space

$\R^n\text{,}$ but it is not so great for helping understand matrix products

$A\uvec{x}$ for a particular matrix

$A\text{.}$

🔗

In Discovery 24.1, we discovered that for an

$n\times n$ matrix

$A\text{,}$ if we can build a basis for

$\R^n$ consisting of eigenvectors of

$A\text{,}$ then every matrix product

$A\uvec{x}$ becomes simple to compute once

$\uvec{x}$ is decomposed as a linear combination of these basis vectors. Indeed, if

$\{\uvec{u}_1,\uvec{u}_2,\dotsc,\uvec{u}_n\}$ is a basis for

$\R^n\text{,}$ and we have

$\begin{align*} A\uvec{u}_1 \amp= \lambda_1\uvec{u}_1, \amp A\uvec{u}_2 \amp= \lambda_2\uvec{u}_2, \amp \amp\dotsc, \amp A\uvec{u}_n \amp= \lambda_n\uvec{u}_n, \end{align*}$

🔗

then multiplication by

$A$ can be achieved by scalar multiplication:

$\begin{align*} \amp\amp \uvec{x} \amp= k_1\uvec{u}_1 + k_2\uvec{u}_2 + \dotsb + k_n\uvec{u}_n\\ \\ \amp\implies \amp A\uvec{x} \amp= k_1A\uvec{u}_1 + k_2A\uvec{u}_2 + \dotsb + k_nA\uvec{u}_n\\ \amp\amp\amp= k_1\lambda_1\uvec{u}_1 + k_2\lambda_2\uvec{u}_2 + \dotsb + k_n\lambda_n\uvec{u}_n. \end{align*}$

🔗

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.