Properties

The basic matrix operations are addition, scalar multiplication, and vector multiplication. We assume the reader is familiar with these. In addition, we will refer to the following unary operations on square matrices:

Inversion: The inverse $(\mathsf{A}^{-1}){}^a{}_b$ of a matrix $\mathsf{A}^a{}_b$ is one such that their matrix product is the identity matrix $\bd ^a{}_b$ (whose elements $\delta^i{}_j$ are just the Krönecker delta function).

Transposition: The transpose $(\mathsf{A}^T){}^a{}_b$ of a matrix $\mathsf{A}^a{}_b$ is given by interchanging the indecies on each component: $(A^T){}^i{}_j=A^j{}_i$.

Conjugation: The Hermitian conjugate (adjoint) $(\mathsf{A}^\dag ){}^a{}_b$ of a matrix $\mathsf{A}^a{}_b$ is given by interchanging the indecies and taking the complex conjugate of each component: $(A^\dag ){}^i{}_j={A^j{}_i}^*$.

A matrix that is identical to its own transpose is called symmetric. A matrix whose transpose is identical to the original matrix's inverse is called orthogonal. A matrix that is identical to its own Hermitian conjugate is called Hermitian (or self-adjoint. A matrix whose Hermitian conjugate is identical to the original matrix's inverse is called unitary.

At present, the routines under this header only deal with real matrices (i.e. matrices, vectors, and scalars whose components are all real). In this case, symmetric is equivalent to Hermitian, and orthogonal is equivalent to unitary.