Prove A' is the Generalized Inverse of A When A'A is Idempotent

  • Thread starter siucw
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In summary, the conversation discusses the concept of a generalized inverse of a matrix A being equal to A' if A'A is idempotent. This means that A'A is equal to its own transpose. The conversation also delves into the properties of projection and the relationship between the column space of AA' and the rank of A'. It is noted that the column space of AA' is a subspace of A and that AA' is a projection onto this subspace. The conversation also mentions the importance of proving equations and using concepts such as projections to show the relationship between A and A'.
  • #1
siucw
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generalized inverse of A is equal to A' if A'A is idempotent?
 
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  • #2
A'A idempotent means A'AA'A=A'A, don't know what you mean by ' though - if A and A' are invertible then you can cancel and get AA'=I but you don't know A and A' are invertible do you?
 
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  • #3
A' is transpose of A,
A'A idempotent should mean A'AA'A=A'A, right?
 
  • #4
Yes, realized and corrected myself.

A'A is thus symmetric and idempotent... exactly what do you mean by "generalized" inverse?
 
  • #5
AA-A=A,A- is generalized inverse, and no assumption about full rank or not for A
 
  • #6
All I've figured is that:

A'AA'A=A'A => A'(AA'A - A) =0

So it boils down to showing this implies AA'A - A = 0

which must use the fact that A' is the transpose, some how... can't quite explain it, sorry.
 
  • #7
If you're trying to show A' is the generalized inverse, the symmetric properties are essentially freebies. The hard part is showing AA'A=A and A'AA'=A', but if you've got one you've got the other, so you can concentrate on the first one.

You know AA' is idempotent, so it's a projection, the question is onto what? What can you say about the column space of AA'? What can you say about it's rank? Do you have any guess as to what it should be projecting onto?
 
  • #8
rank A'A is equal to rank A
 
  • #9
Good, now what about the column space of AA' (or it's image if you prefer to think of it that way)?
 
  • #10
it is a subspace of A
 
  • #11
so, you can prove it conceptually: since AA' is a projection, AA'A is projecting A to a space which include AA', then...?
is there any way to show it equationally?
 
  • #12
A is a matrix not a subspace. I think you left out the words 'column space'? You should also be able to say something stronger than 'subspace' at this point-remember the bit about the ranks.

A general thing about projections- if T is a projection onto a subspace U, and v is any vector in U then T(v)=v. You should be able to prove this. You should then be able to prove (AA')v=v where v is any column of A.
 
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1. What is the definition of a generalized inverse?

A generalized inverse of a matrix A is a matrix A' that satisfies the conditions A'A = A' and AA'A = A.

2. Why is it important to prove that A' is the generalized inverse of A when A'A is idempotent?

Proving that A' is the generalized inverse of A when A'A is idempotent ensures that A' can be used to solve equations involving A, which is crucial in many fields of science and engineering.

3. What does it mean for a matrix to be idempotent?

A matrix A is idempotent if A*A = A, meaning that multiplying the matrix by itself once results in the same matrix.

4. How do you prove that A' is the generalized inverse of A when A'A is idempotent?

To prove that A' is the generalized inverse of A when A'A is idempotent, we must show that A'A = A' and AA'A = A. This can be done by multiplying A' and A, and then using the properties of idempotent matrices to simplify the equations.

5. Can A' still be the generalized inverse of A if A'A is not idempotent?

No, A' cannot be the generalized inverse of A if A'A is not idempotent. In order for A' to be the generalized inverse, it must satisfy the conditions A'A = A' and AA'A = A, which can only be true if A'A is idempotent.

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