Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Transposed Matrix

Transposed Matrix The transpose of a given matrix M of order mn is the matrix MT, which is obtained by exchanging the order of the indices: (mrs)T = (msr ). This new matrix MT is of the order nm.

More simply expressed, we just write the rows as columns, and vice versa. This results in a mirroring along the 45 line (which is equivalent to a mirroring of the columns and a consecutive counterclockwise rotation). Here is an example:

It is evident that MTT equals M, where MTT is the transpose of the transpose of M.
Symmetric Matrix The matrix M is called symmetric if M= MT. Example:
Skew-Symmetric Matrix If M = -MT, the matrix is called skew-symmetric or anti-symmetric. The elements of the main diagonal of a anti-symmetric matrix are always zero. Example: