Using symmetry to look at the transpose

One way to define the transpose is to find the main diagonal of a matrix and transpose the matrix elements across it. Alternatively, we can know nothing about the process of finding the transpose and define the transpose from the definition of symmetric and antisymmetric matrices.

symmetric matrix = a matrix which equals its own transpose
antisymmetric matrix = a matrix which equals minus its own transpose

All square matrices can be broken down into the sum of a symmetric and an antisymmetric matrix (we need the fact that 2 transposes equals the identity). This means that if we have a group of matrices under addition which includes all symmetric and antisymmetric matrices, then we know that that group contains all square matrices.

I think we should spend more time investigating groups that are closed under transpose.