Hermitian Matrix Algebra
In mathematical physics, a matrix is Hermitian if its eigenvalues are real and its diagonal elements are their complex conjugate. Hermitian matrices are a subspace of the vector space of complex matrices.
Hermitian matrices also share the property that their inverse is a Hermitian matrix. This is called the principle of self-adjointness.
What is a Hermitian Matrix?
A matrix is a collection of quantities, like numbers or symbols, arranged in a tabular form of rows and columns. A Hermitian matrix is a real symmetric matrix that contains complex numbers in the off-diagonal elements.
A Hermitian matrix is positive semi-definite if the determinant and inversion are both positive. Its diagonal elements are always real. A Hermitian matrix can be represented as the product of another Hermitian matrix with its conjugate transpose.
Hermitian matrices are named after Charles Hermite, who showed in 1855 that matrices with this property have the same properties as real symmetric matrices, such as being able to be diagonalized.
A Hermitian matrix has a symmetric pair of eigenvalues and eigenvectors that span the entire complex plane. Its eigenvalues are real, not necessarily positive, and its eigenvectors are complex. Moreover, if a Hermitian matrix is a square, its principal minors are all positive. This is a very useful property of Hermitian matrices.
A matrix is Hermitian if and only if its determinant, inverse, and conjugate are Hermitian. Moreover, the entries on the diagonal of Hermitian matrices are real numbers. Also, they have an orthonormal complete set of eigenvectors (or eigenfunctions).
Another important property of Hermitian matrices is that their eigenvalues are all real. This is due to the Cauchy interlace theorem. The eigenvalues of the leading principal submatrices of Hermitian positive-definite matrices are equal.
As a result, the eigenvalues of Hermitian matrices can be found by solving a system of linear equations. However, it is important to note that this condition only applies when the matrix has a nonzero real scalar in the middle. Otherwise, the matrix could be skew-Hermitian. If this is the case, the commutator of two Hermitian matrices is skew-Hermitian. Thus, if you want to find the eigenvalues of a skew-Hermitian matrix, you must first convert it to a Hermitian one. This process is called normalizing.
In matrix algebra, the trace of a matrix is defined as the sum of the elements on its main diagonal. It is also sometimes referred to as the adjoint or inverse of the matrix. The trace of a Hermitian matrix is zero because all the entries along its main diagonal are real numbers.
A Hermitian matrix has a complex determinant because all the non-diagonal entries are complex numbers. The determinant of a Hermitian matrix is independent of the basis on which it is written, and this property allows the matrix to be rotated or scaled without affecting its determinant.
A matrix with a complex determinant is Hermitian in the sense that any rotation or scaling of the matrix will preserve its determinant. This property is important in physics because it means that the vectors associated with a Hermitian matrix will remain parallel to each other if they are translated or rotated. This is important because it allows the vectors to be used to measure properties of the system.
The eigenvectors of a Hermitian matrix have unique indices and are normalized so that their 2-norm is 1. If a Hermitian operator has distinct left and right eigenvalues, then the corresponding eigenvectors form a complete basis in the vector space where the operator acts. This is known as the Schur-Horn theorem and also the Atiyah-Sternberg theorem in symplectic geometry.
The main diagonal entries in a Hermitian matrix are always real numbers. This is a very important property because it means that the matrix can be used to represent physical systems, such as a quantum mechanical system, where all of the observables are represented by the real values of the components.
Another important property of Hermitian matrices is that the algebraic multiplicity of each eigenvector is equal to its geometric multiplicity. This is a very important property because this implies that the matrix is diagonalizable and has k linearly independent eigenvectors. It is also useful to know that Hermitian matrices always have an orthonormal set of eigenvectors.