**Hermitian Matrices:**Many Hamiltonians have this property especially those containing magnetic fields: where at least some elements are complex.**Real Symmetric Matrices:**These are the commonest matrices in physics as most Hamiltonians can be represented this way: and all are real. This is a special case of Hermitian matrices.**Positive Definite Matrices:**A special sort of Hermitian matrix in which all the eigenvalues are positive. The overlap matrices used in*tight-binding*electronic structure calculations are like this. Sometimes matrices are*non-negative definite*and zero eigenvalues are also allowed. An example is the dynamical matrix describing vibrations of the atoms of a molecule or crystal, where .**Unitary Matrices:**The product of the matrix and its Hermitian conjugate is a unit matrix, . A matrix whose columns are the eigenvectors of a Hermitian matrix is unitary; the unitarity is a consequence of the orthogonality of the eigenvectors. A scattering () matrix is unitary; in this case a consequence of current conservation.**Diagonal Matrices:**All matrix elements are zero except the diagonal elements, when . The matrix of eigenvalues of a matrix has this form. Finding the eigenvalues is equivalent to*diagonalisation*.**Tridiagonal Matrices:**All matrix elements are zero except the diagonal and first off diagonal elements, , . Such matrices often occur in 1 dimensional problems and at an intermediate stage in the process of diagonalisation.**Upper and Lower Triangular Matrices:**In Upper Triangular Matrices all the matrix elements below the diagonal are zero, for . A Lower Triangular Matrix is the other way round, for . These occur at an intermediate stage in solving systems of equations and inverting matrices.**Sparse Matrices:**Matrices in which most of the elements are zero according to some pattern. In general sparsity is only useful if the number of non-zero matrix elements of an matrix is proportional to rather than . In this case it may be possible to write a function which will multiply a given vector by the matrix to give without ever having to store all the elements of . Such matrices commonly occur as a result of simple discretisation of partial differential equations, and in simple models to describe many physical phenomena.**General Matrices:**Any matrix which doesn't fit into any of the above categories, especially non-square matrices.

**Complex Symmetric Matrices:**Not generally a useful symmetry. There are however two related situations in which these occur in theoretical physics: Green's functions and scattering () matrices. In both these cases the real and imaginary parts commute with each other, but this is not true for a general complex symmetric matrix.**Symplectic Matrices:**This designation is used in 2 distinct situations:- The eigenvalues occur in pairs which are reciprocals of one another. A common example is a Transfer Matrix.
- Matrices whose elements are
*Quaternions*, which are matrices like

Such matrices describe systems involving spin-orbit coupling. All eigenvalues are doubly degenerate (*Kramers degeneracy*).