# Some basic questions regarding rank-1 matrices

Okay, we can show this in different ways, depending on how we define rank. I'll try to show these using both these definitions:

• $$\operatorname{rank}(A)$$ is the dimension of the range of $$A$$.
• $$\operatorname{rank}(A)$$ is the least number $$r$$ of rank 1 matrices $$A_1, A_2, \dots, A_r$$ so that $$A = A_1 + A_2 + \dots + A_r$$ and a rank one matrix $$A_1$$ is defined as a matrix that can be written as an outer product: $$A_1 = xy^T$$ for vectors $$x, y$$.

These two definitions are equivalent. The proof of this is left as an exercise to the reader.

## If $$B$$ has rank 1, then $$AB$$ has at most rank one

Fixed the formulation for you on this one. $$AB$$ can have rank zero.

## Definition 1

$$B$$ has rank one, so its range is one dimensional. It follows that its nullspace is $$n-1$$-dimensional. Consider the range of $$AB$$. If $$x$$ is in the nullspace of $$B$$, then $$ABx = A0 = 0$$, so $$AB$$'s nullspace is also at least $$n-1$$-dimensional, so its range is at most 1-dimensional.

Note that for $$AB$$ to have rank 1 you must have that $$Ax \neq 0$$ for some $$x$$ in the range of $$B$$. A sufficient, but not necessary, condition for this is that $$A$$ is invertible.

## Definition 2

$$B$$ has rank one, so it can be written $$B = xy^T$$. Then $$AB = Axy^T = (Ax)y^T$$, so $$AB$$ can be written as one outer product (but we don't know if it can be written using zero outer products, which would be the case if $$Ax = 0$$).

## Definition 1

As said before, if $$B$$ has rank 1, then its nullspace is $$n-1$$ dimensional. Pick a basis $$v_1, \dots, v_{n-1}$$ for the nullspace of $$B$$. These are eigenvectors of $$B$$ with eigenvalue zero.

## Definition 2

Say $$B = xy^T$$. Then if you multiply a vector $$v$$ with $$B$$ you get $$xy^Tv$$. Note that $$y^Tv$$ is the inner product of $$y$$ and $$v$$. The orthogonal complement $$\mathcal U$$ to the subspace spanned by $$y$$ has dimension $$n-1$$. If $$u \in \mathcal U$$ then $$y^Tu = 0$$ and hence $$Bu = xy^Tu = x \cdot 0 = 0$$ so if you pick a basis for $$\mathcal U$$, this basis will be eigenvectors to $$B$$ with eigenvalue 0.

## Is there a non-zero eigenvalue?

In short, it is not guaranteed that there will be a non-zero eigenvalue.

This rank one nilpotent matrix: $$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$ has zero as eigenvalue with algebraic multiplicity 2 and geometric multiplicity 1. It therefore has $$n-1$$ zero eigenvalues, but it does not have 1 non-zero eigenvalue. In other words, it is not guaranteed that you will have a non-zero eigenvalue for rank one matrices.

It comes from the associavity of matrix multiplication. If $B$ has rank-1 then it can be written in the form $B = u v^T$ for some vectors $u$ and $v$. So, $$AB = A (uv^T) = (Au) v^T$$ but $Au$ is a vector itself, so now we have a rank-1 expression for $AB$. Maybe it will be helpful to see a picture of the shapes of the matrices during this process:

$$\underbrace{\begin{bmatrix}\cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot\end{bmatrix}}_{A} \left(\underbrace{\begin{bmatrix}\cdot \\ \cdot \\ \cdot\end{bmatrix}}_{u}\underbrace{\begin{bmatrix}\cdot & \cdot & \cdot\end{bmatrix}}_{v^T}\right)= \left(\underbrace{\begin{bmatrix}\cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot\end{bmatrix}}_{A}\underbrace{\begin{bmatrix}\cdot \\ \cdot \\ \cdot\end{bmatrix}}_{u}\right)\underbrace{\begin{bmatrix}\cdot & \cdot & \cdot\end{bmatrix}}_{v^T} = \underbrace{\begin{bmatrix}\cdot \\ \cdot \\ \cdot\end{bmatrix}}_{Au}\underbrace{\begin{bmatrix}\cdot & \cdot & \cdot\end{bmatrix}}_{v^T}$$

(Of course, note that if $Au=0$, then the result is the zero matrix so the rank would be zero instead of 1)

The rank of a matrix is the dimension of its image ${\rm rank} \ B = \dim \{Bx: x \in \mathbb R^n\},$ or equivalently the dimension of the space spanned by its columns.

Since applying $AB$ on $\mathbb R^n$ is the same as applying $A$ on the one-dimensional space $B\mathbb R^n$, we see that $AB$ can only have at most rank 1. Note that rank 0 is possible when the columns of $B$ are orthogonal to the rows of $A$.

We have the rank-nullity theorem $n = {\rm rank} \ B + \dim \ker B$ (i.e. what doesn't contribute to the dimensions of the image must land in the kernel.

If ${\rm rank} \ B = 1$, then the dimension of the kernel must be $n-1$, i.e. the eigenspace for the eigenvalue is $n-1$. We can deduce that there is also another nonzero eigenvalue since otherwise we'd have ${\rm rank} \ AB = 0$.