Inner products

  • Thread starter physicsss
  • Start date
319
0
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
 

robphy

Science Advisor
Homework Helper
Insights Author
Gold Member
5,381
661
Can you use the determinant?
 
319
0
Yes, but how does that help me?
 

matt grime

Science Advisor
Homework Helper
9,394
3
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.
 
694
0
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.
 

matt grime

Science Advisor
Homework Helper
9,394
3
snap. why was the title inner products though?
 

Related Threads for: Inner products

  • Posted
Replies
2
Views
1K
  • Posted
Replies
7
Views
2K
  • Posted
Replies
10
Views
3K
  • Posted
Replies
3
Views
2K
  • Posted
Replies
4
Views
617
  • Posted
Replies
4
Views
2K
  • Posted
Replies
1
Views
2K
  • Posted
Replies
2
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top