We have often explored new ideas in Linear Algebra by making connections to our previous algebraic experience. Adding two numbers, \(x+y\text{,}\) led us to adding vectors \(\vx+\vy\) and adding matrices \(\tta+\ttb\text{.}\) We explored multiplication, which then led us to solving the matrix equation \(\ttaxb\text{,}\) which was reminiscent of solving the algebra equation \(ax=b\text{.}\)
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, \(x=0\text{,}\) we know that in some sense \(5x\) will be “5 times bigger than \(x\text{.}\)” Applying this to vectors, we would readily agree that \(5\vx\) gives a vector that is “5 times bigger than \(\vx\)”; we know from Part 8 of Theorem 2.2.16 that \(\norm{5\vx} = 5\norm{\vx}\text{.}\)
Within the linear algebra context, though, we have two types of multiplication: scalar and matrix multiplication. What happens to \(\vx\) when we multiply it by a matrix \(\tta\text{?}\) 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 \(\tta\vx = a\vx\text{,}\) where \(a\) is some scalar? That is the motivating question of this chapter.
