Proving Symmetry of A + A(Transpose) for Any Square Matrix A

In summary, to show that A + A(transpose) is symmetric for any square matrix A, we must start with the definitions of AT and symmetric matrix. The definition of a symmetric matrix is that A = AT, not A = A(transpose) as previously stated. Therefore, A + A(transpose) = A + A = A (since A = AT), proving that A + A(transpose) is symmetric for any square matrix A.
  • #1
HKP
4
0

Homework Statement


Show that A + A(transpose) is symmetric for any square matrix A.

Homework Equations


A symmetric matrice is equal to A=A(transpose)


The Attempt at a Solution



I said A + A(transpose)
= A(transpose) + A(transpose)
= (A + A) (transpose)
= A + A

I don't know what else to do
 
Physics news on Phys.org
  • #2
It looks like you're assuming that A is symmetric right at the start, which isn't information that you're given.

Start with the definitions of AT and symmetric matrix.

The definition you show for a symmetric matrix isn't right - I think you want this:
A symmetric matrix A is such that A = AT. (A matrix can't be "equal" to an equation.)
 

Related to Proving Symmetry of A + A(Transpose) for Any Square Matrix A

1. How do you prove symmetry of A + A(Transpose) for any square matrix A?

To prove symmetry of A + A(Transpose), we need to show that (A + A(Transpose))(Transpose) = A + A(Transpose). This can be done by expanding the left side of the equation and showing that it is equal to the right side. This proves that the matrix is symmetric.

2. Why is it important to prove symmetry of a matrix?

Proving symmetry of a matrix is important because it provides information about the matrix's properties and structure. Symmetric matrices have many useful properties and are often easier to work with in calculations and applications.

3. Can you provide an example of a symmetric matrix?

Yes, an example of a symmetric matrix is the identity matrix, which has 1s along the main diagonal and 0s elsewhere. Another example is the matrix [1 2; 2 3], where the elements on either side of the main diagonal are equal.

4. Is every square matrix symmetric?

No, not every square matrix is symmetric. A matrix is considered symmetric if it is equal to its transpose. Therefore, a matrix that is not equal to its transpose is not symmetric.

5. Are there any other methods for proving symmetry of a matrix?

Yes, there are other methods for proving symmetry of a matrix, such as using the properties of transposition and proving that the matrix is equal to its negative. However, the method of showing that (A + A(Transpose))(Transpose) = A + A(Transpose) is the most commonly used and straightforward method for proving symmetry.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
17
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
867
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top