How does v * vT equal a matrix?

Question

I don't understand how a vector * vector.T results in a matrix? Shouldn't the result be a single product?

For instance

multiplied with its transposed form.

Maybe I am confused as to how to multiply a vector with a transposed vector.

There is a huge difference between $v\cdot v^T$ and $v^T\cdot v$. — Arthur, Jun 23 '18 at 12:54

score 2 · Answer 1 · answered Jun 23 '18 at 12:56

When you multiply a $3\times1$ vector by a $1\times3$ vector the result is $(3\times1)\cdot(1\times3)=3\times3$.

On the other hand, when you multiply a $1\times3$ vector by a $3\times1$ vector, you get $(1\times3)\cdot(3\times1)=1\times1$ (this is the inner product, aka the dot product)

The image in this answer is the easiest way to visualize this, imo.

score 1 · Answer 2 · answered Jun 23 '18 at 12:56

In this example you are thinking of a vector in $\mathbb{R}^3$ as a "row vector" - that is, as a $1 \times 3$ matrix. Its transpose is a column vector - a $3\times 1$ matrix.

Multiplying a $1 \times 3$ by a $3 \times 1$ matrix produces a $1 \times 1$ matrix. Sometimes it's useful to think of the dot product this way, sometimes just as a scalar (the single entry in that matrix), sometimes not.

How does v * vT equal a matrix?

2 Answers2