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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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Creator
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  • 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. – Marc van Leeuwen Apr 23 '15 at 07:29
  • @MarcvanLeeuwen I think OP means the outer product of a vector with itself. – Travis Willse Apr 23 '15 at 07:30
  • @Travis excellent, would mind to have a look at my other question and .. any comment would be highy appreciated. http://math.stackexchange.com/questions/1247598/understanding-eigenvector – Creator Apr 23 '15 at 07:47
  • It appears I misinterpreted outer product as cross product, or vector product. Deleting my answer – Marc van Leeuwen Apr 23 '15 at 07:50
  • @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? – Travis Willse Apr 23 '15 at 08:06

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

Travis Willse
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  • 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 – Fabio Apr 10 '18 at 07:47
  • @Fabio how does this generalize to projecting $y$ onto the subspace spanned by the columns of a matrix? I posted a question here – Euler_Salter Jun 22 '21 at 09:29