Question

Question 1: Geometry of the SVD (a) A k-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points (, R, 2k satisfying (1. Note that we can have a kedimenslonal ellipse embedded inside IRA even in the case n>k by allowing some of the zi to be identically zero. Using these deinitions, show that the matrix Σ diag(ơi, ,omin (m,n) e Rmxn, where σ12 σ12 in 2 Ơmin (rn,n)2 0, maps the unit -phere {x e Rn : llzlla 1), surface and interior, to an ellipse. Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the n and n > m separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce r s min (m, m) such

Answer 1

onal ellipse, Suv foce and lnlerior oith nes olo A K- dimens the Standard Co odinates alobcty defned os the set of points z=(z, za zk at considex 4he dinensional ettpse IR" given by Nouw E is given b

Let ac f-> unit SPnae o Ev, the -dimensional ellipsoid given by be a Polợt on the SU face 아 Uラ Let久EU for TA to be on the Surface of E."n must satis to the suface of E

mapred to the Soyace 이 Er 2. 2. Mi n E.

E R let 2 eu be a foint on he 9ujace i. edrx ỉ9 on the suǐ foce ol the elliese) only

5 ơa ''s on the Sv-face of the etipse