Inner Product and Orthogonal Complement of Symmetric and Skew-Symmetric Matrices

[ahamdiheme]

Homework Statement

Consider the vector space \Re^nxn over \Re, let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y\in\Re^nxn define their inner product by <X,Y>=Tr(X^TY). Show that, with respect to this inner product,
R=S^\bot

Homework Equations

Definition of inner product
Definition of orthogonal compliment
Definition of symmetric matrix
Definition of skew symmetric matrix

The Attempt at a Solution

If i can show that
R-S^\bot=0
will it be sufficient and how do i go about it?

 

[HallsofIvy]

What do you mean by R- S^{\bot}= 0? To show that R= S^{\bot} you must show that the inner product of any member of R with any member of S is 0, that's all.