Let U be an m m matrix, V be an n n matrix, and let where 1 is an n n diagonal

Let U be an m × m matrix, V be an n × n matrix, and let

where ˆ‘1 is an n × n diagonal matrix with diagonal entries σ1, σ2,..., σn and O is the (m - n) × n zero matrix.
(a) If U = (U1, U2), where U1 has n columns, show that
Uˆ‘ = U1ˆ‘1
(b) Show that if A = Uˆ‘VT then A can be expressed as an outer product expansion of the form