Let be a set of real matrices such that
, where is the identity matrix;
if and ,then either or , but not both;
if and ,then either or ;
if and ,there is at least one such that .
Is contains at most matrices?
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The orthogonal group of order n has two components. One which includes the identity transformation (sometimes called the special rotation group) and the other component includes a reflection. When represented by real n × n matrices, the first group is has a determinant of 1 and the other component has a determinant of -1 (the reflection). The size of the first component is 2 n ( n + 1 ) and of the other component 2 n ( n − 1 ) ] . The sum of these two triangular numbers is n 2 .