Any square matrix over a field is similar to its transpose and any square complex matrix is similar to a symmetric complex matrix. We investigate the situation for real orthogonal (respectively complex unitary) similarity.

Is the transpose of a matrix similar to the original matrix?

With a square, symmetric matrix, the transpose of the matrix is the original matrix. A correlation matrix will always be a square, symmetric matrix so the transpose will equal the original.

What matrices are called similar?

Two square matrices are said to be similar if they represent the same linear operator under different bases. Two similar matrices have the same rank, trace, determinant and eigenvalues.