Find a basis for the set of all the matrices in the form of: $\begin{bmatrix} p & -q & -r & -s \\ q & p & s & -r \\ r & -s & p & q \\ s & r & -q & p \end{bmatrix}$. Then, check that if $p^2 + q^2 + r^2 + s^2 = 1$, the matrix is orthogonal.
Is the basis not just the column vectors? There's no relation between them and the other ones. Also, I'm so confused about the second part. Can anyone walk me through this?
On
Consider what you get from $p=1,q=0,r=0,s=0$ and the similar possibilities: $$ \mathbb{1},\quad I=\begin{bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix},\quad J=\begin{bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix},\quad K=\begin{bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} $$ so your matrices have the form $p\mathbb{1}+qI+rJ+sK$ (where $\mathbb{1}$ is the identity matrix).
Why do they form a basis?
What are the transpose of these matrices?
Can you compute $IJ$, $JI$, $IK$, $KI$, $JK$ and $KJ$? How doed this help in proving thee last statement?
Hint:
The matrix can be written as
$$pA+qB+rC+sD$$
for some matrices $A,B,C,D$.
To check a matrix, $Q$ is orthogonal, compute $Q^TQ$ and verify that it is an identity matrix.