Question

Problems 1-2 compute the SVD of a square singular matrix A. 1. Compute ATA and its eigenvalues o, 0 and unit eigenvectors V1, V2: 1 4 A = 2 8 2. (a) Compute AAT and its eigenvalues o, 0 and unit eigenvectors un, u2. (b) Choose signs so that Avi = 01u1 and verify the SVD: 1 4 0 1 U2 V1 28 (c) Which four vectors give orthonormal bases for C(A), N(A), C(AT), N(AT)? U1 o] [ vz]".

I'm looking for specific insight on 2c. I'm not sure how to identify these vectors at all based on what I've calculated so far. Thank you!

Answer 1

sigg 2 c) we have SvD of A in 1 61 . u, Ug 으로 2 8 185 0 2 1/35 2/15 -2155 1155 8 o o "155 IST Now, suppose we have Α! ΟΣΥ ui Er ų mx (m-r met O 94 X nel axr 0 Emerar nancy om-) (91-4 [4 2x T Vi mxy mx (nl) T V₂ nuxn प mxu Zr XX VT xxn (Full Rank singular vame Decompositions) here, m=2, n=2, r=1 Then, A- ui U2. 2X Er IXI EXT 0 Thus, here, u forms a basis for CCA) basil you NCAT) a basis for CCAT) V2 forms . Uz youmea . Vi basis for N(A)

. They, we get the four fundamental subspaces 1/15 C(A) = 2155 - 2/5 NCA)= WH CCATJE "JA 47119 NCAT) - -2 / vis 1/15