# Let V be the vector space of real 2 x 2 matrices with inner product (A|B) = tr(B^tA). Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for U^perp where U^{perp}left{A in V |(A|B)=0 forall B in U right}

Question
Matrices
Let V be the vector space of real 2 x 2 matrices with inner product
(A|B) = tr(B^tA).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for $$U^\perp$$ where $$U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}$$

2021-01-18
Step 1
To find an orthogonal basis of $$U^\perp$$ , where U is the subspace consisting of symmetric matrices.
Let $$A \in U^\perp$$ then
$$tr(B^tA)=0$$
$$\Rightarrow tr(B^{t}A)=-tr(B^{t}A)$$
$$=-tr(BA), \text{ since B is a symmetric matrix }$$
$$=tr(-AB)$$
Hence A is a skew symmetric matrix.
Step 2
Now the basis of $$U^\perp$$ is $$\left\{\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix}0 & -1 \\ -1 & 0 \end{bmatrix} , \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}$$
$$tr\left(\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix}0 & -1 \\ -1 & 0 \end{bmatrix} \right)= tr \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=0$$
$$tr\left(\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix} , \begin{bmatrix}0 & -1 \\ -1 & 0 \end{bmatrix} \right)= tr \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=0$$
$$tr\left(\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix} \right)= tr \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}=0$$
Hence the given basis is an orthogonal basis.

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