Hermitian matrices

From SklogWiki
Jump to: navigation, search

A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex elements which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:

a_{i,j} = a_{j,i}^*.

If the conjugate transpose of a matrix A is denoted by A^\dagger, then this can concisely be written as

  A = A^\dagger. \,

For example,

   \begin{bmatrix}3&2+i\\ 2-i&1\end{bmatrix}

All eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. The typical example of a Hermitian matrix in physics is the Hamiltonian (specially in quantum mechanics).

References[edit]