# Properties of this rank-$1$ matrix

I've forgotten a lot of basic linear algebra so I was hoping someone might be able to provide a few properties of this matrix.

$$\mathbf b \mathbf a^\top = \begin{pmatrix} a_1b_1 & a_2b_1 & a_3b_1 & \cdots & a_nb_1 \\ a_1b_2 & a_2b_2 & a_3b_2 & \cdots & a_nb_2 \\ a_1b_3 & a_2b_3 & a_3b_3 & \cdots & a_nb_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1b_n & a_2b_n & a_3b_n & \cdots & a_nb_n \end{pmatrix}$$

I know the determinant is $$0$$ and the rank is $$1$$, so the nullity must be $$n-1$$. Are there any other properties this matrix has?

Thanks in advance.

• By the way, the row $i$ and column $j$ entry is $b_ia_j$, so the matrix equals $\mathbf{b}\mathbf{a}^T$, where $\mathbf{a} = \begin{pmatrix}a_1 \\ a_2 \\ \vdots \\ a_n\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}b_1 \\ b_2 \\ \vdots \\ b_n\end{pmatrix}$. Maybe you can search online for something like "properties of matrices of the form $\mathbf{u}\mathbf{v}^T$" to find out more about them (e.g. their eigenvalues and eigenvectors). Sep 22, 2019 at 10:08
• Sep 22, 2019 at 10:13
• Also look up “outer product.”
– amd
Sep 23, 2019 at 0:49

## 1 Answer

For example: $$det(I + ba^T) = 1 + a^Tb$$