0

I'm a little conflicted as to why we consider $\vec{a}^T\vec{b}$ to be the same as dot product $\vec{a}\cdot\vec{b}$ is some calculations, where both $\vec{a}$ and $\vec{b}$ of let's say size $n\times1$.

We know that $\vec{a}\vec{b}^T$ results in a matrix of size $n \times n$, shouldn't $\vec{a}^T\vec{b}$ result in a matrix of size $1 \times 1$? In comparison, $\vec{a}\cdot\vec{b}$ is a scalar.

I feel like I'm missing some kind of convention?

Darkhan
  • 111
  • 3
    $1\times 1$ matrices are interchangeable with real numbers in practice, just as the integer $1$ is interchangeable with the real number $1$ or the rational number $1$ or the complex number $1$, etc... Yes, they may formally be different objects, but they share enough properties that we will fluidly swap between them without explaining that we are in practice. – JMoravitz Mar 10 '21 at 17:23
  • A matrix of size $1\times 1$ is just a single scalar. – Berci Mar 10 '21 at 17:30
  • @JMoravitz I see, so a scalar can also be represented as $1 \times 1$ matrix, or $1 \times 1 \times 1$ matrix, or $1 \times 1 \times 1 \times 1 \ldots$ matrix? – Darkhan Mar 10 '21 at 17:35
  • If you have the time, I thoroughly recommend reading When is one thing equal to another thing? by Dr. Barry Mazur, professor from Harvard. It is a bit long, but well explains the habits of mathematicians and our lax attitude when it comes to calling certain things "equal" when they happen to be formally defined in different ways. – JMoravitz Mar 10 '21 at 17:45
  • That said, I personally think of it as that a $1\times 1$ matrix or a $1\times 1\times 1$ matrix and so on can be thought of as a scalar in their corresponding scalar field, be it $\Bbb R^{1\times 1}$ or $\Bbb R^{1\times 1\times 1}$ and so on, and recognizing that those scalar fields have the same properties as our usual scalar field $\Bbb R$ and that the matrix $[x]$ plays the same role and has the same properties in $\Bbb R^{1\times 1}$ as the real number $x$ has in $\Bbb R$. – JMoravitz Mar 10 '21 at 17:51
  • I would not say that it can be represented as the other, but I would say that it can be interpreted or seen as being analogous to, etc... – JMoravitz Mar 10 '21 at 17:53

0 Answers0