# linalg

***

**1. Solving Systems of Linear Equations**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::VectorXd b(3);
  b << 10, 11, 12;
  Eigen::VectorXd x = A.lu().solve(b);
  std::cout << "Solution: " << x << "\n";
  return 0;
}
```

**2. Computing Eigenvalues and Eigenvectors**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::EigenSolver<Eigen::MatrixXd> eigensolver(A);
  std::cout << "Eigenvalues:\n" << eigensolver.eigenvalues() << "\n";
  std::cout << "Eigenvectors:\n" << eigensolver.eigenvectors() << "\n";
  return 0;
}
```

**3. Calculating Matrix Norm**

```cpp
#include <Eigen/Norm>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  std::cout << "Matrix norm: " << A.norm() << "\n";
  return 0;
}
```

**4. Finding Matrix Inverse**

```cpp
#include <Eigen/LU>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::MatrixXd Ainv = A.inverse();
  std::cout << "Matrix inverse:\n" << Ainv << "\n";
  return 0;
}
```

**5. Generating Random Matrices**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A = Eigen::MatrixXd::Random(3, 3);
  std::cout << "Random matrix:\n" << A << "\n";
  return 0;
}
```

**6. Solving Least Squares Problems**

```cpp
#include <Eigen/QR>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::VectorXd b(3);
  b << 10, 11, 12;
  Eigen::VectorXd x = A.colPivHouseholderQr().solve(b);
  std::cout << "Least squares solution: " << x << "\n";
  return 0;
}
```

**7. Calculating Matrix Determinant**

```cpp
#include <Eigen/Determinant>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  std::cout << "Matrix determinant: " << A.determinant() << "\n";
  return 0;
}
```

**8. Generating Random Vectors**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::VectorXd v = Eigen::VectorXd::Random(3);
  std::cout << "Random vector:\n" << v << "\n";
  return 0;
}
```

**9. Performing Matrix Multiplication**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::MatrixXd B(3, 3);
  B << 10, 11, 12, 13, 14, 15, 16, 17, 18;
  Eigen::MatrixXd C = A * B;
  std::cout << "Matrix multiplication:\n" << C << "\n";
  return 0;
}
```

**10. Calculating Matrix Trace**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  std::cout << "Matrix trace: " << A.trace() << "\n";
  return 0;
}
```

**11. Generating Random Matrices with Singular Values**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A = Eigen::MatrixXd::Random(3, 3);
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A, Eigen::ComputeFullU | Eigen::ComputeFullV);
  std::cout << "Singular values:\n" << svd.singularValues() << "\n";
  return 0;
}
```

**12. Eigenvalue Decomposition for Real Symmetric Matrices**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> eigensolver(A);
  std::cout << "Eigenvalues:\n" << eigensolver.eigenvalues() << "\n";
  std::cout << "Eigenvectors:\n" << eigensolver.eigenvectors() << "\n";
  return 0;
}
```

**13. Eigensystem Analysis for Complex Matrices**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXcd A(3, 3);
  A << 1 + 2i, 3 - 4i, -5, 3 - 4i, 4 + 5i, 1 + 2i, -5, 1 + 2i, 6;
  Eigen::ComplexEigenSolver<Eigen::MatrixXcd> eigensolver(A);
  std::cout << "Eigenvalues:\n" << eigensolver.eigenvalues() << "\n";
  std::cout << "Eigenvectors:\n" << eigensolver.eigenvectors() << "\n";
  return 0;
}
```

**14. Cholesky Decomposition for Positive Definite Matrices**

```cpp
#include <Eigen/Cholesky>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::LLT<Eigen::MatrixXd> chol(A);
  std::cout << "Cholesky decomposition:\n" << chol.matrixL() << "\n";
  return 0;
}
```

**15. QR Decomposition for General Matrices**

```cpp
#include <Eigen/QR>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::HouseholderQR<Eigen::MatrixXd> qr(A);
  std::cout << "QR decomposition:\nQ:\n" << qr.householderQ() << "\nR:\n" << qr.matrixQR().triangularView<Eigen::Upper>() << "\n";
  return 0;
}
```

**16. Solving Linear Least Squares with QR Decomposition**

```cpp
#include <Eigen/QR>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::VectorXd b(3);
  b << 1, 2, 3;
  Eigen::HouseholderQR<Eigen::MatrixXd> qr(A);
  Eigen::VectorXd x = qr.solve(b);
  std::cout << "Solution to linear least squares:\n" << x << "\n";
  return 0;
}
```

**17. Generating Random Orthogonal Matrices**

```cpp
#include <Eigen/Orthogonal>

int main() {
  Eigen::MatrixXd A = Eigen::MatrixXd::Random(3, 3);
  Eigen::HouseholderQR<Eigen::MatrixXd> qr(A);
  std::cout << "Random orthogonal matrix:\n" << qr.householderQ() << "\n";
  return 0;
}
```

**18. Singular Value Decomposition (SVD)**

```cpp
#include <Eigen/SVD>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A, Eigen::ComputeFullU | Eigen::ComputeFullV);
  std::cout << "SVD:\nU:\n" << svd.matrixU() << "\nSigma:\n" << svd.singularValues().asDiagonal() << "\nV:\n" << svd.matrixV() << "\n";
  return 0;
}
```

**19. Calculating Matrix Rank**

```cpp
#include <Eigen/SVD>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A, Eigen::ComputeFullU | Eigen::ComputeFullV);
  std::cout << "Matrix rank: " << svd.rank() << "\n";
  return 0;
}
```

**20. Computing Matrix Condition Number**

```cpp
#include <Eigen/SVD>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A, Eigen::ComputeFullU | Eigen::ComputeFullV);
  std::cout << "Matrix condition number: " << svd.singularValues()(0) / svd.singularValues()(svd.singularValues().size() - 1) << "\n";
  return 0;
}
```

**21. Generating Random Unit Vectors**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::VectorXd v = Eigen::VectorXd::Random(3).normalized();
  std::cout << "Random unit vector:\n" << v << "\n";
  return 0;
}
```

**22. Finding Vector Norm**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::VectorXd v(3);
  v << 1, 2, 3;
  std::cout << "Vector norm: " << v.norm() << "\n";
  return 0;
}
```

**23. Calculating Vector Dot Product**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::VectorXd v1(3), v2(3);
  v1 << 1, 2, 3;
  v2 << 4, 5, 6;
  std::cout << "Vector dot product: " << v1.dot(v2) << "\n";
  return 0;
}
```

**24. Calculating Vector Cross Product**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::Vector3d v1(1, 2, 3), v2(4, 5, 6);
  std::cout << "Vector cross product:\n" << v1.cross(v2) << "\n";
  return 0;
}
```

**25. Spherical Interpolation of 3D Vectors**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::Vector3d v1(1, 2, 3), v2(4, 5, 6);
  double alpha = 0.5;
  std::cout << "Spherical interpolation:\n" << v1 * std::sin((1 - alpha) * M_PI) + v2 * std::sin(alpha * M_PI) << "\n";
  return 0;
}
```

**26. Generating Random Quaternions**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q = Eigen::Quaterniond::UnitRandom();
  std::cout << "Random quaternion:\n" << q.coeffs() << "\n";
  return 0;
}
```

**27. Quaternion Multiplication**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q1(1, 2, 3, 4), q2(5, 6, 7, 8);
  std::cout << "Quaternion multiplication:\n" << q1 * q2 << "\n";
  return 0;
}
```

**28. Quaternion Conjugation**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q(1, 2, 3, 4);
  std::cout << "Quaternion conjugation:\n" << q.conjugate() << "\n";
  return 0;
}
```

**29. Quaternion Inverse**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q(1, 2, 3, 4);
  std::cout << "Quaternion inverse:\n" << q.inverse() << "\n";
  return 0;
}
```

**30. Quaternion Rotation Matrix**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q(1, 2, 3, 4);
  std::cout << "Quaternion rotation matrix:\n" << q.toRotationMatrix() << "\n";
  return 0;
}
```

**31. Quaternion Slerp Interpolation**

```cpp
#include <Eigen/Quaternion>

int main() {
  Eigen::Quaterniond q1(1, 2, 3, 4), q2(5, 6, 7, 8);
  double alpha = 0.5;
  std::cout << "Quaternion Slerp interpolation:\n" << q1.slerp(alpha, q2).coeffs() << "\n";
  return 0;
}
```

**32. Solving Linear Systems with Sparse Matrices**

```cpp
#include <Eigen/Sparse>

int main() {
  Eigen::SparseMatrix<double> A(3, 3);
  A.insert(0, 0) = 1;
  A.insert(0, 1) = 2;
  A.insert(0, 2) = 3;
  A.insert(1, 0) = 4;
  A.insert(1, 1) = 5;
  A.insert(1, 2) = 6;
  A.insert(2, 0) = 7;
  A.insert(2, 1) = 8;
  A.insert(2, 2) = 9;
  Eigen::VectorXd b(3);
  b << 10, 11, 12;
  Eigen::VectorXd x = A.llt().solve(b);
  std::cout << "Solution to linear system with sparse matrix:\n" << x << "\n";
  return 0;
}
```

**33. Iterative Refinement for Linear Systems**

```cpp
#include <Eigen/IterativeLinearSolvers>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 4, 5, 6, 7, 8, 9;
  Eigen::VectorXd b(3);
  b << 10, 11, 12;
  Eigen::VectorXd x0 = Eigen::VectorXd::Zero(3);
  Eigen::ConjugateGradient<Eigen::MatrixXd> solver;
  solver.setTolerance(1e-6);
  Eigen::VectorXd x = solver.solveWithGuess(A, b, x0);
  std::cout << "Solution to linear system with iterative refinement:\n" << x << "\n";
  return 0;
}
```

**34. Solving Lyapunov Equations**

```cpp
#include <Eigen/Cholesky>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::MatrixXd Q(3, 3);
  Q << 10, 11, 12, 13, 14, 15, 16, 17, 18;
  Eigen::MatrixXd X = A.llt().solve(Q);
  std::cout << "Solution to Lyapunov equation:\n" << X << "\n";
  return 0;
}
```

**35. Computing Sparse Matrix Eigenvalues**

```cpp
#include <Eigen/SparseEigen>

int main() {
  Eigen::SparseMatrix<double> A(3, 3);
  A.insert(0, 0) = 1;
  A.insert(0, 1) = 2;
  A.insert(0, 2) = 3;
  A.insert(1, 0) = 4;
  A.insert(1, 1) = 5;
  A.insert(1, 2) = 6;
  A.insert(2, 0) = 7;
  A.insert(2, 1) = 8;
  A.insert(2, 2) = 9;
  Eigen::GeneralizedSelfAdjointEigenSolver<Eigen::SparseMatrix<double>> eigensolver(A);
  std::cout << "Eigenvalues of sparse matrix:\n" << eigensolver.eigenvalues() << "\n";
  return 0;
}
```

**36. Performing Matrix Decomposition**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::EigenSolver<Eigen::MatrixXd> eigensolver(A);
  std::cout << "Eigenvalues:\n" << eigensolver.eigenvalues() << "\n";
  std::cout << "Eigenvectors:\n" << eigensolver.eigenvectors() << "\n";
  Eigen::MatrixXd V = eigensolver.eigenvectors();
  Eigen::MatrixXd D = eigensolver.eigenvalues().asDiagonal();
  std::cout << "A = V * D * V^{-1}:\n" << V * D * V.inverse() << "\n";
  return 0;
}
```

**37. Reducing Matrix Dimensionality**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  int k = 2;
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Eigen::MatrixXd U = svd.matrixU();
  Eigen::VectorXd S = svd.singularValues();
  Eigen::MatrixXd V = svd.matrixV();
  Eigen::MatrixXd A_reduced = U.leftCols(k) * S(Eigen::seqN(0, k)).asDiagonal() * V.topRows(k).adjoint();
  std::cout << "Reduced matrix:\n" << A_reduced << "\n";
  return 0;
}
```

**38. Computing Matrix Exponentials**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::MatrixExponential<Eigen::MatrixXd> expA(A);
  std::cout << "Matrix exponential:\n" << expA << "\n";
  return 0;
}
```

**39. Calculating Matrix Trace**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  std::cout << "Matrix trace: " << A.trace() << "\n";
  return 0;
}
```

**40. Finding Linear Combinations**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 4);
  A << 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12;
  Eigen::VectorXd b(3);
  b << 1, 2, 3;
  Eigen::LeastSquaresConjugateGradient<Eigen::MatrixXd> solver;
  Eigen::VectorXd x = solver.solve(A, b);
  std::cout << "Linear combination solution:\n" << x << "\n";
  return 0;
}
```

**41. Computing Matrix Pseudoinverse**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd A(3, 4);
  A << 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12;
  Eigen::MatrixXd pinvA = A.completeOrthogonalDecomposition().pseudoInverse();
  std::cout << "Matrix pseudoinverse:\n" << pinvA << "\n";
  return 0;
}
```

**42. Orthogonal Linear Regression**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd X(3, 5);
  X << 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15;
  Eigen::VectorXd y(3);
  y << 1, 2, 3;
  Eigen::VectorXd beta = (X.transpose() * X).ldlt().solve(X.transpose() * y);
  std::cout << "Orthogonal linear regression coefficients:\n" << beta << "\n";
  return 0;
}
```

**43. Principal Component Analysis (PCA)**

```cpp
#include <Eigen/SVD>

int main() {
  Eigen::MatrixXd data(10, 3);
  data << 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30;
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(data, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Eigen::MatrixXd U = svd.matrixU();
  std::cout << "PCA components:\n" << U << "\n";
  return 0;
}
```

**44. Image Compression**

```cpp
#include <Eigen/Dense>

int main() {
  Eigen::MatrixXd image(512, 512);
  // Load image data...
  Eigen::JacobiSVD<Eigen::MatrixXd> svd(image, Eigen::ComputeFullU | Eigen::ComputeFullV);
  int k = 100;
  Eigen::MatrixXd U = svd.matrixU().leftCols(k);
  Eigen::MatrixXd S = svd.singularValues().head(k).asDiagonal();
  Eigen::MatrixXd V = svd.matrixV().topRows(k).adjoint();
  Eigen::MatrixXd compressedImage = U * S * V;
  // Save compressed image...
  return 0;
}
```

**45. Finite Difference Stencils**

```cpp
#include <Eigen/Sparse>

int main() {
  int n = 100;
  Eigen::SparseMatrix<double> stencil(n, n);
  for (int i = 1; i < n - 1; i++) {
    for (int j = 1; j < n - 1; j++) {
      stencil.insert(i, j) = -4;
      stencil.insert(i, j - 1) = 1;
      stencil.insert(i, j + 1) = 1;
      stencil.insert(i - 1, j) = 1;
      stencil.insert(i + 1, j) = 1;
      stencil.makeCompressed();
    }
  }
  // Use stencil for solving Poisson's equation...
  return 0;
}
```

**46. Sparse Matrix Operations**

```cpp
#include <Eigen/Sparse>

int main() {
  Eigen::SparseMatrix<double> A(3, 3);
  A.insert(0, 0) = 1;
  A.insert(0, 1) = 2;
  A.insert(0, 2) = 3;
  A.makeCompressed();
  Eigen::VectorXd b(3);
  b << 10, 11, 12;
  Eigen::VectorXd x = A.colPivHouseholderQr().solve(b);
  std::cout << "Solution to sparse linear system:\n" << x << "\n";
  return 0;
}
```

**47. Sparse Matrix Factorization**

```cpp
#include <Eigen/Sparse>

int main() {
  Eigen::SparseMatrix<double> A(3, 3);
  A.insert(0, 0) = 1;
  A.insert(0, 1) = 2;
  A.insert(0, 2) = 3;
  A.insert(1, 0) = 4;
  A.insert(1, 1) = 5;
  A.insert(1, 2) = 6;
  A.makeCompressed();
  Eigen::SparseLU<Eigen::SparseMatrix<double>> lu(A);
  std::cout << "Sparse LU factorization:\n" << lu.matrixL() << "\n" << lu.matrixU() << "\n";
  return 0;
}
```

**48. Eigenvalue Analysis for Symmetric Positive Definite Matrices**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> eigensolver(A);
  std::cout << "Eigenvalues of symmetric positive definite matrix:\n" << eigensolver.eigenvalues() << "\n";
  return 0;
}
```

**49. Matrix Function Evaluation**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::MatrixFunction<Eigen::MatrixXd> expA(std::exp);
  std::cout << "Matrix exponential:\n" << expA(A) << "\n";
  return 0;
}
```

**50. Eigenvalue Filtering**

```cpp
#include <Eigen/Eigenvalues>

int main() {
  Eigen::MatrixXd A(3, 3);
  A << 1, 2, 3, 2, 4, 5, 3, 5, 6;
  Eigen::EigenSolver<Eigen::MatrixXd> eigensolver(A);
  Eigen::VectorXd eigenvalues = eigensolver.eigenvalues();
  Eigen::VectorXd filteredEigenvalues = eigenvalues.cwiseAbs() > 1;
  std::cout << "Filtered eigenvalues:\n" << filteredEigenvalues << "\n";
  return 0;
}
```