Types of Matrices

There are several ways of classifying matrices depending on symmetry, sparsity etc. Here we provide a list of types of matrices and the situation in which they may arise in physics.

• Hermitian Matrices: Many Hamiltonians have this property especially those containing magnetic fields: $A_{ji} = A^\dagger_{ij}$ 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: $A_{ji} = A_{ij}$ and all $A_{ij}$ 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 $\omega^2 \ge 0$.
• Unitary Matrices: The product of the matrix and its Hermitian conjugate is a unit matrix, $\bss{U}^\dagger\bss{U} = \bss{I}$. 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 ($\bss{S}$) matrix is unitary; in this case a consequence of current conservation.
• Diagonal Matrices: All matrix elements are zero except the diagonal elements, $A_{ij} = 0$ when $i\not= j$. 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, $A_{i,i}\not=0$, $A_{i,i\pm 1}\not=0$. 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, $A_{ij} = 0$ for $i>j$. A Lower Triangular Matrix is the other way round, $A_{ij} = 0$ for $i<j$. 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 $N\times N$ matrix is proportional to $N$ rather than $N^2$. In this case it may be possible to write a function which will multiply a given vector $\bi{x}$ by the matrix $\bss{A}$ to give $\bss{A}\bi{x}$ without ever having to store all the elements of $\bss{A}$. 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.

There are a few extra types which arise less often:

• 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 ($\bss{S}$) 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 $2\times 2$ matrices like
    

    $$\left(\begin{array}{ll}a&b\\ -b^*&a^*\end{array}\right).$$ (3.1)

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