Rank One Matrices
Suppose we have two vectors $\mathbf u \in \mathbb R^m$ and $\mathbf v \in \mathbb R^n$. Then multiplication $\mathbf u \times \mathbf v^T$ gives us a matrix $A = \mathbf u \cdot \mathbf v^T$, $A \in \mathbb R^{m \times n}$
- This multiplication produces rank-1 matrices
Rank 1 Matrices
$\mathbf u \times \mathbf v^T = \begin{bmatrix} a \ b \ c \end{bmatrix} \big[1 \ 2 \big] = \begin{bmatrix} 1a & 2a \ 1b & 2b \ 1c & 2c \end{bmatrix}$
Subspaces
This matrix $A$ is a special matrix:
- all these rows lie on the same line
- all these columns are same directions
- Row Space: all combinations of $\mathbf v$
- Column Space: all combinations of $\mathbf u$
== Projection Matrices == Suppose we want to project to a line $\mathbf u$
-
then the Projection Matrix $P$ is $P = \cfrac{\mathbf u \mathbf u^T}{| \mathbf u|^2} = \mathbf u \mathbf u^T$ - if $\mathbf u$ is a unit vector, e.g. $| \mathbf u |^2 = 1$ - then the projection matrix is just an outer product of $\mathbf u$ with itself