The set of all real matrices forms a real Lie subgroup.

In the case of the real symmetric matrix, we see that , so clearly holds.

The same is true of any a symmetric real matrix.

In particular if is a real symmetric matrix, they are the same except for transposition.

To see this, recall that the eigenvalues of a real matrix may in general be complex.

See the article on 2 x 2 real matrices for other representations of complex numbers.

The most obvious example is the real skew-symmetric matrix.

Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero.

At the end, the participant is asked to indicate on a real matrix where the little man that he or she visualized finished.

Let be defined on , where is a real, positive definite matrix.