Symmetric matrices satisfy
ST=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 AH=A are called ``Hermitian matrices.'' They are all over quantum mechanics.
Skew-symmetric matrices satisfy
KT= -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 dij=0 if .Upper triangular matrices have only nonzero elements on and above the main diagonal:
In index notation, uij=0 if j<i.Lower triangular matrices:
In index notation, lij=0 if j>i.The transpose of an upper triangular matrix is a lower triangular one and vice-versa.