TITLE: Geometric meaning of outer product of a vector with itself
QUESTION [5 upvotes]: This question is related to the question in the link below:
Is there a geometric meaning to the outer product of two vectors?
The answer is clear, but I am wondering: If we take a outer product of a vector with itself, then is there a specific geometric meaning of the matrix which is not evident from an interpretation of the outer product of two general vectors?

REPLY [8 votes]: If $\bf x$ is a unit vector, then the linear transformation
$${\bf y} \mapsto ({\bf x}{\bf x}^T)({\bf y}) = ({\bf x} \cdot {\bf y}) \bf x$$ defined by the outer product ${\bf x}{\bf x}^T$ is the orthogonal projection of $\bf y$ onto the line spanned by $\bf x$. For general $\bf x$, the linear transformation is this projection composed with dilation by a factor ${\bf x} \cdot {\bf x} = ||{\bf x}||^2$.