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}

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
asked 2021-01-17
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 \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)

Answers (1)

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.
0

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