# Symmetric matrix

• Apr 16th 2010, 02:26 AM
vuze88
Symmetric matrix
If $A$ is a square matrix and $x^TAy=(Ax)^Ty$ for all $x,y\in R^n$, then prove $A$ is a symmetric matrix. ie. prove $A^T=A$

I cant just go $x^TAy=x^TA^Ty$ rite? So what should i do instead?
• Apr 17th 2010, 12:26 AM
BobP
Use the matrix property $(AB)^T=B^TA^T.$
• Apr 17th 2010, 02:01 AM
vuze88
yeh i know that property but how do i do it? ive just used the property and now im at $x^TAy=x^TA^Ty$. But i cant just cancel them out to get $A=A^T$ because they are matrices...
• Apr 17th 2010, 04:08 AM
BobP
You're told that the relationship is true for all $x$ and $y$, so it's certainly true for specific $x$ and $y.$
Choose $x$ to consist of entirely of zeros apart from a 1 in the ith position and $y$ to consist entirely of zeros apart from 1 in the jth position. Then evaluate both sides of the equation $x^TA^Ty=x^TAy.$
• Apr 18th 2010, 12:11 AM
Noxide
You started with:
A is nxn
x^TAy= (Ax)^Ty
so
x^TAy= x^TA^Ty
x^TAy - x^TA^Ty = 0
factor...
x^T(A - A^T)y = 0
this is true for all x,y in R^n
consider x = 0, and y = anything in R^n we get x^T(A - A^T)y = 0
consider y = 0, and x = anything in R^n we get x^T(A - A^T)y = 0
consider y=x=0 we get x^T(A - A^T)y = 0
we know the statement is true for all x,y not just x, y, being zero vectors
the only way x^T(A - A^T)y = 0 is true for any x and y is if A- A^T = zero matrix

hope this helped