User:IssaRice/Linear algebra/Singular value decomposition

the stupid textbooks don't tell you anything about SVD!!!! i think it's super helpful to look at all the wrong things one might say about SVD... we need to un-knot all those wrong intuitions. i'll list some knots that i have had.

starting at this image: https://en.wikipedia.org/wiki/File:Singular-Value-Decomposition.svg

• if A is an invertible matrix, then $A=E_{1}\cdots E_{m}$ for some elementary matrices $E_{1},\ldots ,E_{m}$. Dilations and swapping elementary matrices obviously involve only orthogonal operations. So we can write A as an alternating product of orthogonal and shear matrices (the product of two orthogonal matrices is again orthogonal. right???). If we can prove SVD for shears, we can convert this to an alternating product of orthogonal and diagonal matrices. unfortunately, this doesn't seem to lead to a full proof of SVD (unless orthogonal and diagonal matrices somehow commute).
• one question one might have is, to get the behavior of M in the image, can't we just squish along the standard basis directions, then rotate? surely this would produce the same ellipse. And it would seem that we've only required one rotation, instead of the two in SVD. That's true, but pay attention to where the basis vectors went. A squish followed by a rotation... would preserve orthogonality. But in M it is clear that these basis vectors are no longer orthogonal. So even though we have faithfully preserved the ellipse, we don't have the same transformation. i.e. $M(\{v:\|v\|=1\})=M'(\{v:\|v\|=1\})$ need not imply $M=M'$, apparently.