Unitary Matrix / Helmite Matrix

Unitary Matrix

When a square matrix is as described below, it is called a normal matrix.

A^{\dagger}A=AA^{\dagger}=I

When A is a normal matrix, there exists a unitary matrix that can diagonalize A as follows

U^{\dagger}AU = \sum_j \lambda_j \ket{j}\bra{j}

This means that there is normal orthogonal basis.

A = \sum_j e^{i\theta_j} \ket{\psi_j}\bra{\psi_j}

Since Unitary matrix is normal matrix, Unitary matrix should be decomposed as

U = \sum_j e^{i\theta_j} \ket{\psi_j}\bra{\psi_j}

The helmite matrix is special case of matrix, when the Hermitian conjugate (adjoint matrix) is equal to the original matrix.

H^{\dagger}=H

So, its eigen values take real values in this case

H = \sum_j \lambda_j \ket{\psi_j}\bra{\psi_j}

Using a Hermitian operator ensures that a physical quantity is derived as a real number.