TITLE: How can you explain the Singular Value Decomposition to non-specialists?
QUESTION [39 upvotes]: In two days, I am giving a presentation about a search engine I have been making the past summer. My research involved the use of singular value decompositions, i.e., $A = U \Sigma V^T$. I took a high school course on Linear Algebra last year, but the course was not very thorough, and though I know how to find the SVD of a matrix, I don't know how to explain what I have in my hands after the matrix has been decomposed.
To someone who has taken linear algebra, I can say that I can decompose a matrix $A$ into matrix  $\Sigma$, whose diagonal holds the singular values, and matrices $U$ and $V$ whose columns represent the left and right singular vectors of matrix $A$. I am not sure how to explain what a singular value or what left/right singular vectors are. I can still be satisfied if there is no easy way to explain what this decomposition means, but I always prefer keeping the audience as informed as possible.

REPLY [3 votes]: My layman's explanation is to consider the reverse mapping in 2 dimensions:

Consider any 2 independent vectors lying in a plane. both starting at the origin. Rotate the vectors by an arbitrary angle (a) so that the vectors lie above and below the x-axis, and importantly such that the vector with the larger x value has a lower absolute y value than the other. The 2 points will lie on the boundary of an ellipse centered on the origin.
Scale the ellipse (and 2 vectors) to a unit circle and inspect the angle between the two (unit) vectors. 
Vary the angle a of the rotation and repeat the scaling until the scaled unit vectors are orthogonal.
Finally rotate the unit vectors by angle b until the correspond to i and j.

a,b and the scaling parameters define the SVD (actually the inverse).