Struct GeneralizedSchurDecomposition 

pub struct GeneralizedSchurDecomposition<R: Runtime> {
    pub q: Tensor<R>,
    pub z: Tensor<R>,
    pub s: Tensor<R>,
    pub t: Tensor<R>,
    pub eigenvalues_real: Tensor<R>,
    pub eigenvalues_imag: Tensor<R>,
}

Generalized Schur (QZ) decomposition result: A = Q @ S @ Z^H, B = Q @ T @ Z^H

For a pair of matrices (A, B), the QZ decomposition factors them as:

• A = Q @ S @ Z^H
• B = Q @ T @ Z^H

where Q and Z are unitary, S is upper quasi-triangular, and T is upper triangular.

Generalized Eigenvalues

The generalized eigenvalues λ satisfy det(A - λB) = 0 and are computed as the ratios of diagonal elements: λ_i = S[i,i] / T[i,i] (when T[i,i] ≠ 0).

When T[i,i] = 0, the eigenvalue is infinite.

Use Cases

• Generalized eigenvalue problems: Av = λBv
• Control theory: descriptor systems
• Stability analysis of matrix pencils

Fields

q: Tensor<R>

Left unitary matrix Q [n, n]

z: Tensor<R>

Right unitary matrix Z [n, n]

s: Tensor<R>

Upper quasi-triangular matrix S (from A) [n, n]

t: Tensor<R>

Upper triangular matrix T (from B) [n, n]

eigenvalues_real: Tensor<R>

Real parts of generalized eigenvalues [n]

eigenvalues_imag: Tensor<R>

Imaginary parts of generalized eigenvalues [n]