type Vec = readonly number[] | Float64Array;
/**
 * Generalized eigenvalues of the pencil `A x = λ B x` (B nonsingular), via the
 * eigendecomposition of `B⁻¹A` — reusing `inv`, `multiply`, and the corrected
 * `eigs` (Hessenberg + Francis double-shift). Eigenvalues match
 * `scipy.linalg.eig(A, B)` across real, complex-pair and clustered spectra. For
 * a *singular* or numerically near-singular `B` this squaring-free QZ formulation
 * breaks down; extracting the pencil eigenvalues directly from the {@link qz}
 * factors (`diag AA / diag BB`, with 2×2-block handling for complex pairs) is the
 * future enhancement for that regime.
 */
export declare function generalizedEig(A: readonly number[][], B: readonly number[][]): {
    values: Array<number | {
        re: number;
        im: number;
    }>;
    eigenvectors?: unknown;
};
/** Lower-triangular part of `A`, zeroing entries above the `k`-th diagonal. */
export declare function tril(A: readonly number[][], k?: number): number[][];
/** Upper-triangular part of `A`, zeroing entries below the `k`-th diagonal. */
export declare function triu(A: readonly number[][], k?: number): number[][];
/**
 * Vandermonde matrix of `x`: column `j` is `xᵢ` to a power. With `increasing`
 * false (default, NumPy convention) column `j` is `xᵢ^(N-1-j)`; `N` defaults to
 * `x.length`.
 */
export declare function vander(x: Vec, n?: number, increasing?: boolean): number[][];
/**
 * Toeplitz matrix: first column `c`, first row `r` (defaults to `c`). Constant
 * along each diagonal. `T[i][j] = c[i-j]` for `i ≥ j`, else `r[j-i]`.
 */
export declare function toeplitz(c: Vec, r?: Vec): number[][];
/** Circulant matrix: each row is the previous one rotated right by one. */
export declare function circulant(c: Vec): number[][];
/**
 * Companion matrix of a monic-normalized polynomial given by coefficients
 * `[a₀, a₁, …, a_n]` (highest degree first, NumPy `np.companion` convention).
 * Its eigenvalues are the polynomial's roots.
 */
export declare function companion(coeffs: Vec): number[][];
/**
 * Graph Laplacian of an adjacency matrix (bridge graph ↔ linear algebra — spectral
 * graph theory). Combinatorial `L = D − A` by default (D = diagonal degree matrix);
 * with `{ normalized: true }`, the symmetric normalized Laplacian
 * `L_sym = I − D^(−1/2) A D^(−1/2)`. Eigen-analysis of `L` (via `eigs`) gives the
 * Fiedler vector / spectral clustering.
 */
export declare function laplacianMatrix(adjacency: readonly number[][], opts?: {
    normalized?: boolean;
}): number[][];
/**
 * Natural log of the absolute determinant via LU (the `logabsdet` component of
 * `numpy.linalg.slogdet`), stable where `log(det(A))` would overflow. Returns
 * `{ sign, value }` with `det = sign · exp(value)`.
 */
export declare function logdet(A: readonly number[][]): {
    sign: number;
    value: number;
};
/**
 * Generalized (QZ) Schur decomposition of the pencil `(A, B)` with `B` nonsingular:
 * returns orthogonal `Q`, `Z` and upper-(quasi-)triangular `AA`, `BB` with
 * `A = Q·AA·Zᵀ` and `B = Q·BB·Zᵀ`. Built from the real Schur of `B⁻¹A` (= Z S Zᵀ) and
 * the QR of `B·Z` (= Q·BB): then `AA = Qᵀ·A·Z`. The Schur step is the hardened
 * Hessenberg + Francis double-shift `matrixSchur` (see {@link realSchur}), so `qz`
 * no longer stalls on non-symmetric `B⁻¹A` pencils; matches the decomposition
 * contract of `scipy.linalg.qz` (the factors are not unique).
 */
export declare function qz(A: readonly number[][], B: readonly number[][]): {
    AA: number[][];
    BB: number[][];
    Q: number[][];
    Z: number[][];
};
export {};
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