Special matrices

Symmetric matrices satisfy

S^T=S

Symmetric matrices are very common in engineering. For example, most statics deals with symmetric matrices, as does solid body dynamics, and a lot of the simpler fluid flows.

Complex matrices for which A^H=A are called ``Hermitian matrices.'' They are all over quantum mechanics.

Skew-symmetric matrices satisfy

K^T= -K

Skew-symmetric matrices determine the velocity field in solid body motion, and other fields involving cross products.

Example: the following is a skew symmetric matrix:

Diagonal matrices have only nonzero elements on the main diagonal:

An example is the unit matrix. In index notation, a matrix is diagonal iff d_ij=0 if .

Upper triangular matrices have only nonzero elements on and above the main diagonal:

In index notation, u_ij=0 if j<i.

Lower triangular matrices:

In index notation, l_ij=0 if j>i.

The transpose of an upper triangular matrix is a lower triangular one and vice-versa.