- #1
s_j_sawyer
- 21
- 0
Hello everyone. This is my first official post here but I have been lurking around for about a year now.
Prove that a matrix A is symmetric if and only if x*Ay = Ax*y for all x,y of R^n, where * denotes the dot product.
So I was able to do prove the forward direction, as follows:
Assume A is symmetric. Then
x*Ay = x^T A y
= x^T A^T y
= (Ax)^T y
= Ax * y as required.
However, I am completely stumped for the other direction. I.e., assuming x*Ay = Ax*y and then showing A = A^T.
Any suggestions?
Homework Statement
Prove that a matrix A is symmetric if and only if x*Ay = Ax*y for all x,y of R^n, where * denotes the dot product.
Homework Equations
The Attempt at a Solution
So I was able to do prove the forward direction, as follows:
Assume A is symmetric. Then
x*Ay = x^T A y
= x^T A^T y
= (Ax)^T y
= Ax * y as required.
However, I am completely stumped for the other direction. I.e., assuming x*Ay = Ax*y and then showing A = A^T.
Any suggestions?