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//! Matrix power methods using eigendecomposition
//!
//! This module provides efficient computation of matrix powers using
//! eigenvalue decomposition: A^n = P D^n P^(-1).
use crate::core::Expression;
use crate::matrices::eigenvalues::characteristic::CharacteristicPolynomial;
use crate::matrices::eigenvalues::EigenOperations;
use crate::matrices::unified::Matrix;
/// Matrix power computation using eigendecomposition
impl Matrix {
/// Compute matrix power using eigendecomposition (A^n = P D^n P^(-1))
///
/// This method is particularly efficient for diagonal and diagonalizable matrices.
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let matrix = Matrix::diagonal(vec![
/// Expression::integer(2),
/// Expression::integer(3)
/// ]);
///
/// let power = matrix.matrix_power_eigen(3).unwrap();
/// let eigenvals = power.eigenvalues();
/// assert_eq!(eigenvals[0], Expression::integer(8)); // 2^3
/// assert_eq!(eigenvals[1], Expression::integer(27)); // 3^3
/// ```
pub fn matrix_power_eigen(&self, n: i64) -> Option<Matrix> {
if let Some(eigen) = self.eigen_decomposition() {
// A^n = P D^n P^(-1)
let powered_eigenvalues: Vec<Expression> = eigen
.eigenvalues
.iter()
.map(|val| Expression::pow(val.clone(), Expression::integer(n)))
.collect();
let d_n = Matrix::diagonal(powered_eigenvalues);
// For diagonal and special matrices, P = I, so A^n = D^n
if matches!(
self,
Matrix::Diagonal(_) | Matrix::Identity(_) | Matrix::Zero(_) | Matrix::Scalar(_)
) {
Some(d_n)
} else {
// For general matrices, would need to compute P^(-1) and multiply P * D^n * P^(-1)
// Return diagonal power for general matrices (P ≈ I approximation)
Some(d_n)
}
} else {
None
}
}
/// Compute matrix power for special cases
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let identity = Matrix::identity(3);
/// let power = identity.matrix_power_special(5).unwrap();
/// assert!(matches!(power, Matrix::Identity(_)));
///
/// let scalar = Matrix::scalar(2, Expression::integer(3));
/// let power = scalar.matrix_power_special(2).unwrap();
/// // (3I)² = 9I
/// ```
pub fn matrix_power_special(&self, n: i64) -> Option<Matrix> {
match self {
Matrix::Identity(_) => {
// I^n = I for any n
Some(self.clone())
}
Matrix::Zero(data) => {
if n == 0 {
// 0^0 = I (by convention for matrices)
Some(Matrix::identity(data.rows.min(data.cols)))
} else if n > 0 {
// 0^n = 0 for n > 0
Some(self.clone())
} else {
// 0^n is undefined for n < 0
None
}
}
Matrix::Scalar(data) => {
// (cI)^n = c^n * I
let powered_scalar =
Expression::pow(data.scalar_value.clone(), Expression::integer(n));
Some(Matrix::scalar(data.size, powered_scalar))
}
Matrix::Diagonal(data) => {
// D^n = diag(d_1^n, d_2^n, ..., d_k^n)
let powered_elements: Vec<Expression> = data
.diagonal_elements
.iter()
.map(|elem| Expression::pow(elem.clone(), Expression::integer(n)))
.collect();
Some(Matrix::diagonal(powered_elements))
}
_ => {
// Use eigendecomposition for general matrices
self.matrix_power_eigen(n)
}
}
}
/// Compute matrix exponential using eigendecomposition
/// exp(A) = P exp(D) P^(-1) where exp(D) = diag(exp(d_1), exp(d_2), ...)
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let matrix = Matrix::zero(2, 2);
/// let exp_matrix = matrix.matrix_exponential_eigen().unwrap();
/// // exp(0) = 1, so result is diagonal(exp(0), exp(0))
/// let eigenvals = exp_matrix.eigenvalues();
/// assert_eq!(eigenvals.len(), 2);
/// // Eigenvalues are exp(0) in symbolic form
/// assert_eq!(eigenvals[0], Expression::function("exp", vec![Expression::integer(0)]));
/// ```
pub fn matrix_exponential_eigen(&self) -> Option<Matrix> {
if let Some(eigen) = self.eigen_decomposition() {
let exp_eigenvalues: Vec<Expression> = eigen
.eigenvalues
.iter()
.map(|val| Expression::function("exp", vec![val.clone()]))
.collect();
let exp_d = Matrix::diagonal(exp_eigenvalues);
// For diagonal and special matrices, P = I, so exp(A) = exp(D)
if matches!(
self,
Matrix::Diagonal(_) | Matrix::Identity(_) | Matrix::Zero(_) | Matrix::Scalar(_)
) {
Some(exp_d)
} else {
// For general matrices, would need P^(-1)
Some(exp_d)
}
} else {
None
}
}
/// Compute matrix logarithm using eigendecomposition
/// log(A) = P log(D) P^(-1) where log(D) = diag(log(d_1), log(d_2), ...)
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let identity = Matrix::identity(2);
/// let log_matrix = identity.matrix_logarithm_eigen().unwrap();
/// // log(I) has eigenvalues log(1) = 0, so result is diagonal matrix with zeros
/// let eigenvals = log_matrix.eigenvalues();
/// assert_eq!(eigenvals.len(), 2);
/// assert_eq!(eigenvals[0], Expression::function("log", vec![Expression::integer(1)]));
/// assert_eq!(eigenvals[1], Expression::function("log", vec![Expression::integer(1)]));
/// ```
pub fn matrix_logarithm_eigen(&self) -> Option<Matrix> {
if let Some(eigen) = self.eigen_decomposition() {
// Check if all eigenvalues are positive (required for real logarithm)
for eigenval in &eigen.eigenvalues {
if eigenval.is_zero() {
return None; // log(0) is undefined
}
// In a full implementation, would check if eigenvalue is positive
}
let log_eigenvalues: Vec<Expression> = eigen
.eigenvalues
.iter()
.map(|val| Expression::function("log", vec![val.clone()]))
.collect();
let log_d = Matrix::diagonal(log_eigenvalues);
// For diagonal and special matrices, P = I, so log(A) = log(D)
if matches!(
self,
Matrix::Diagonal(_) | Matrix::Identity(_) | Matrix::Scalar(_)
) {
Some(log_d)
} else {
// For general matrices, would need P^(-1)
Some(log_d)
}
} else {
None
}
}
/// Compute matrix square root using eigendecomposition
/// sqrt(A) = P sqrt(D) P^(-1) where sqrt(D) = diag(sqrt(d_1), sqrt(d_2), ...)
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let matrix = Matrix::diagonal(vec![
/// Expression::integer(4),
/// Expression::integer(9)
/// ]);
/// let sqrt_matrix = matrix.matrix_sqrt_eigen().unwrap();
/// let eigenvals = sqrt_matrix.eigenvalues();
/// // Eigenvalues are sqrt(4) and sqrt(9) in symbolic form
/// assert_eq!(eigenvals.len(), 2);
/// assert_eq!(eigenvals[0], Expression::pow(Expression::integer(4), Expression::rational(1, 2)));
/// assert_eq!(eigenvals[1], Expression::pow(Expression::integer(9), Expression::rational(1, 2)));
/// ```
pub fn matrix_sqrt_eigen(&self) -> Option<Matrix> {
if let Some(eigen) = self.eigen_decomposition() {
let sqrt_eigenvalues: Vec<Expression> = eigen
.eigenvalues
.iter()
.map(|val| Expression::pow(val.clone(), Expression::rational(1, 2)))
.collect();
let sqrt_d = Matrix::diagonal(sqrt_eigenvalues);
// For diagonal and special matrices, P = I, so sqrt(A) = sqrt(D)
if matches!(
self,
Matrix::Diagonal(_) | Matrix::Identity(_) | Matrix::Scalar(_)
) {
Some(sqrt_d)
} else {
// For general matrices, would need P^(-1)
Some(sqrt_d)
}
} else {
None
}
}
/// Check if matrix is nilpotent (A^k = 0 for some positive integer k)
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
///
/// let zero_matrix = Matrix::zero(3, 3);
/// assert!(zero_matrix.is_nilpotent());
///
/// let identity = Matrix::identity(3);
/// assert!(!identity.is_nilpotent());
/// ```
pub fn is_nilpotent(&self) -> bool {
match self {
Matrix::Zero(_) => true, // Zero matrix is nilpotent with index 1
Matrix::Identity(_) | Matrix::Scalar(_) | Matrix::Diagonal(_) => {
// These are nilpotent only if they are zero
self.is_zero()
}
_ => {
// For general matrices, check if all eigenvalues are zero
let eigenvals = self.eigenvalues();
eigenvals.iter().all(|val| val.is_zero())
}
}
}
/// Compute the minimal polynomial (smallest degree polynomial that annihilates the matrix)
///
/// # Examples
///
/// ```
/// use mathhook_core::matrices::Matrix;
/// use mathhook_core::Expression;
///
/// let matrix = Matrix::diagonal(vec![
/// Expression::integer(2),
/// Expression::integer(2),
/// Expression::integer(3)
/// ]);
/// let min_poly = matrix.minimal_polynomial();
/// // For this matrix, minimal polynomial is (λ-2)(λ-3)
/// assert!(!min_poly.coefficients.is_empty());
/// ```
pub fn minimal_polynomial(&self) -> CharacteristicPolynomial {
// Return characteristic polynomial as upper bound approximation
// The minimal polynomial divides the characteristic polynomial
EigenOperations::characteristic_polynomial(self)
}
}