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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.

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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: $A,\Sigma, \tilde \Sigma$
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  • $\begingroup$ can you explain how you got those answers? $\endgroup$ – nawman Apr 4 '16 at 17:25
  • $\begingroup$ @nawman it is a lot to explain. Perhaps if you posted a separate question about one of these categories (null-space, range, rank, or norm), you'll get a thorough explanation. $\endgroup$ – Omnomnomnom Apr 4 '16 at 17:38

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