Inner products

  • Thread starter physicsss
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  • #1
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Suppose that A is a square. Show that A is invertible if and only if A^T*A is invertible.

I know that if A is an m X n matrix, m>=n, and rkA=n, then the n X n matrix A^T*A is invertible, and that rk(A^TA)= rkA, but I'm still not sure how to start the proof...

TIA
 

Answers and Replies

  • #2
robphy
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Can you use the determinant?
 
  • #3
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Yes, but how does that help me?
 
  • #4
matt grime
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M is invertible if and only if det(M) is not zero.

det(M) is the same as det(M^t)

det(MN)=det(M)det(N)

if x and y are real (or complex) numbers and xy=0 then one of x or y is zero.
 
  • #5
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det(A) = det(A^t) and det(AB) = det(A)det(B) (for all matrices A, B of the proper size).

If A is invertible, then det(A) != 0, so that det(A^t) != 0, and therefore det(AA^t) = det(A)det(A^t) != 0, i.e. AA^t is invertible.

The converse is similar.
 
  • #6
matt grime
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snap. why was the title inner products though?
 

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