Properties (range, null space, norm, rank) of RREF, SVD, and Reduced SVD of matrix A

Question

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.

SVD: A=UΣV∗

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.

Answer 1

The answers are as follows:

• Same null-space: $A,Q, \tilde V^*$
• Same range: $A,\tilde U$
• Same rank: $A, Q, \Sigma,\tilde \Sigma, \tilde U$
• Same norm: $A,\Sigma, \tilde \Sigma$