If is a square matrix and for all , then prove is a symmetric matrix. ie. prove
I cant just go rite? So what should i do instead?
You're told that the relationship is true for all and , so it's certainly true for specific and
Choose to consist of entirely of zeros apart from a 1 in the ith position and to consist entirely of zeros apart from 1 in the jth position. Then evaluate both sides of the equation
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