trace(A*A) is always positive?

  • Thread starter zeebek
  • Start date
  • Tags
    Positive
In summary, the conversation discusses the statement that for any n x n non-trivial matrix A, trace(A*A) is always positive. There is initially a question about the truth of this statement, but it is later discovered that there is a counter-example. The conversation then goes on to explain the proof that A*A is positive semidefinite and has non-negative eigenvalues. Finally, it is clarified that A*A does indeed follow this property.
  • #1
zeebek
27
0
I have a feeling that for any n x n non-trivial matrix A, trace(A*A) is always positive.
Is it true?
 
Physics news on Phys.org
  • #2
Nevermind, I found contre example.
 
  • #3
I'm eager to hear your counter-example, because the statement is true.
 
  • #4
Let [tex]A= \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}[/tex]
 
  • #5
@HallsofIvy: is this meant to be a counter-example? With this A, I get A*A=I (the identity matrix), which has trace equal to 2.

My proof: for an arbitrary matrix A, the product A*A is self-adjoint (because (A*A)*=A*A) and positive semidefinite (because [itex](A^*Av,v)=(A^*v,A^*v)=\|A^*v\|^2=\|Av\|^2[/itex]). Hence A*A has an orthonormal basis of eigenvectors, i.e. is diagonalizable with non-negative eigenvalues. The trace is then the sum of the eigenvalues, which is non-negative.
 
  • #6
Ah. I has assumed that "A*A" simply meant A times itself. Otherwise, as you say, there is no "counterexample".
 
  • #7
Now I understand that I was wrong. A*A is indeed positive semidefinite
thanks to everybody
 

1. Why is it important to know if trace(A*A) is always positive?

Knowing whether trace(A*A) is always positive is important because it can provide information about the properties of the matrix A. For example, if trace(A*A) is positive, then A*A must be a positive definite matrix, which has many useful applications in mathematics and science.

2. What is the definition of trace(A*A)?

The trace of a square matrix A is the sum of its diagonal elements. When A is multiplied by itself (A*A), the trace becomes the sum of the squares of the diagonal elements.

3. Is trace(A*A) always positive for any matrix A?

No, trace(A*A) is not always positive for any matrix A. It depends on the properties of the matrix A. However, if A is a square matrix, then trace(A*A) will always be a real number.

4. Can trace(A*A) be negative?

No, trace(A*A) cannot be negative. The sum of squares of real numbers is always positive, so trace(A*A) will always be greater than or equal to zero.

5. How can we use trace(A*A) to determine the eigenvalues of A?

The eigenvalues of a square matrix A can be determined using the equation det(A - λI) = 0, where det() is the determinant function and I is the identity matrix. The sum of the eigenvalues is equal to the trace of A, or trace(A) = λ1 + λ2 + ... + λn. Therefore, if trace(A*A) is always positive, it can be concluded that all eigenvalues of A are also positive.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
794
  • Linear and Abstract Algebra
Replies
2
Views
986
  • Linear and Abstract Algebra
Replies
12
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
708
  • Linear and Abstract Algebra
Replies
1
Views
490
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
8
Views
996
  • Math POTW for Graduate Students
Replies
1
Views
734
  • Linear and Abstract Algebra
Replies
2
Views
351
  • Linear and Abstract Algebra
Replies
6
Views
748
Back
Top