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.