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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?

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  • $\begingroup$ 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. $\endgroup$
    – Marc van Leeuwen
    Commented Apr 23, 2015 at 7:29
  • $\begingroup$ @MarcvanLeeuwen I think OP means the outer product of a vector with itself. $\endgroup$
    – Travis Willse
    Commented Apr 23, 2015 at 7:30
  • $\begingroup$ @Travis excellent, would mind to have a look at my other question and .. any comment would be highy appreciated. math.stackexchange.com/questions/1247598/… $\endgroup$
    – Creator
    Commented Apr 23, 2015 at 7:47
  • $\begingroup$ It appears I misinterpreted outer product as cross product, or vector product. Deleting my answer $\endgroup$
    – Marc van Leeuwen
    Commented Apr 23, 2015 at 7:50
  • $\begingroup$ @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? $\endgroup$
    – Travis Willse
    Commented Apr 23, 2015 at 8:06

1 Answer 1

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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 ${\bf x} \cdot {\bf x} = ||{\bf x}||^2$.

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  • $\begingroup$ I think that the answer is simply "it's an area element" from the geometrical standpoint is the ellipsoid build on the matrix x x^T $\endgroup$
    – Fabio
    Commented Apr 10, 2018 at 7:47
  • $\begingroup$ @Fabio how does this generalize to projecting $y$ onto the subspace spanned by the columns of a matrix? I posted a question here $\endgroup$
    – Euler_Salter
    Commented Jun 22, 2021 at 9:29

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