8. Spectral Theory

Spectral Theory

Hermitian, Unitary, and Normal Matrices

A matrix $\mathbf{A}$ is:

• Hermitian if it is equal to its conjugate transpose: $\mathbf{A} = \mathbf{A}^*$.
• Unitary if its conjugate transpose is its inverse: $\mathbf{A}^* \mathbf{A} = \mathbf{A} \mathbf{A}^* = \mathbf{I}$.
• Normal if it commutes with its conjugate transpose: $\mathbf{A} \mathbf{A}^* = \mathbf{A}^* \mathbf{A}$.

Spectral Theorem for Hermitian Matrices

The spectral theorem states that every Hermitian matrix $\mathbf{A}$ can be diagonalized by a unitary matrix $\mathbf{U}$, i.e.,

$$ \mathbf{A} = \mathbf{U} \mathbf{D} \mathbf{U}^* $$

where $\mathbf{D}$ is a diagonal matrix whose diagonal entries are the eigenvalues of $\mathbf{A}$, and $\mathbf{U}^*$ is the conjugate transpose of $\mathbf{U}$.

Applications to Quadratic Forms and Optimization

The spectral theorem has various applications, including:

• Quadratic Forms: The diagonalization of a Hermitian matrix allows for the simplification of quadratic forms, making it easier to analyze their properties.
• Optimization: In optimization problems, the spectral theorem can be used to find the minimum or maximum values of quadratic functions subject to certain constraints.