Transposes, symmetry and triangular matrices

The transpose of an m x n matrix A is the n x m matrix AT formed by interchanging the rows and columns of A.

Properties of transposes

For any matrices A and B:

ATT = A

If A and B can be added, then (A + B)T = AT + BT

If A and B can be multiplied, then (AB)T = BTAT. (Notice the reversed order!)

There are several special types square matrices you should know about.

A square matrix A is symmetric if it equals its own transpose:
A = AT.

A square matrix A is skew-symmetric if it equals the negative of its transpose:
A = –AT.

(Notice that the entries on the main diagonal must be zeroes.)

A square matrix is upper triangular if all the entries below its main diagonal are 0.

A square matrix is lower triangular if all the entries above its main diagonal are 0.

A square matrix is diagonal if all the entries both above and below its main diagonal are 0.

Properties of triangular and diagonal matrices.

The product of two upper triangular matrices is upper triangular.

The product of two lower triangular matrices is lower triangular.

The product of two diagonal matrices is diagonal, and its diagonal elements are the products of the corresponding diagonal elements of the original matrices.

For any n = 0, 1, 2, ..., the nth power of a diagonal matrix is diagonal, and its diagonal elements are the nth powers of the diagonal elements of the original matrix.