Consider the following three decompositions of an n by m, rank-r, matrix A:
RREF: A = LQ
where Q is the RREF form of A and L is invertible.
where U and V are unitary and Σ only has non-zero entries on the diagonal.
Reduced SVD: A = U ̃Σ ̃V ̃∗
where U ̃ and V ̃ each a have r orthonormal columns, and Σ ̃ is square diagonal.
For each of the following questions, write down all of the matrices, L, Q, U, Σ, V, U ̃ , Σ ̃ , V ̃ , U∗,V∗,U ̃∗,V ̃∗ that have same range/null space/ norm/rank as A.
What I got so far is A is a square matrix because L is invertible and produce the LA = Q = RREF(A). That means all the matrices will be the same size. Also, no matrices should have the same norm?
If someone can tell me the relationships/properties between all the matrices, that will be very helpful.