# Geometric meaning of outer product of a vector with itself

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?

• Do you mean outer product of two vectors (in three dimensional Euclidean space) in your title? I cannot see what an outer product of one vector would be. The outer product of any vector with itself is the zero vector. Commented Apr 23, 2015 at 7:29
• @MarcvanLeeuwen I think OP means the outer product of a vector with itself. Commented Apr 23, 2015 at 7:30
• @Travis excellent, would mind to have a look at my other question and .. any comment would be highy appreciated. math.stackexchange.com/questions/1247598/… Commented Apr 23, 2015 at 7:47
• It appears I misinterpreted outer product as cross product, or vector product. Deleting my answer Commented Apr 23, 2015 at 7:50
• @Creator I edited the title and body of the question for clarity; would you please ensure that I've managed to preserved your intended meaning and revert/modify as necessary? Commented Apr 23, 2015 at 8:06

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 ${\bf x} \cdot {\bf x} = ||{\bf x}||^2$.
• @Fabio how does this generalize to projecting $y$ onto the subspace spanned by the columns of a matrix? I posted a question here Commented Jun 22, 2021 at 9:29