Sylvester Equation Calculator

Category: Linear Algebra
- July 02, 2025
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Solve the Sylvester equation AX + XB = C for matrix X, where A, B, and C are given matrices. This fundamental matrix equation appears in control theory, signal processing, and numerical linear algebra.

Note: The Sylvester equation has a unique solution if and only if A and -B have no common eigenvalues. Matrix dimensions must be compatible: A (m×m), X (m×n), B (n×n), C (m×n).

Matrix Dimensions

Size of square matrix A and number of rows in X and C
Size of square matrix B and number of columns in X and C

Solution Options

Number of decimal places in results

Understanding the Sylvester Equation

The Sylvester equation AX + XB = C is a fundamental matrix equation that appears frequently in control theory, signal processing, and numerical linear algebra. It generalizes the simple linear equation ax + xb = c to matrices.

Mathematical Foundation

Standard Form: AX + XB = C

Vectorized Form: (I ⊗ A + BT ⊗ I)vec(X) = vec(C)

Solvability Condition: λi(A) + λj(-B) ≠ 0 for all i,j

Solution Methods
  • Direct Method: Using Kronecker products to convert to linear system
  • Iterative Methods: Fixed-point iteration and gradient-based approaches
  • Schur Decomposition: Transform to upper triangular form for efficient solution
  • Bartels-Stewart Algorithm: Most efficient method for large matrices
Applications
  • Control Theory: Pole placement and observer design
  • Signal Processing: Filter design and system identification
  • Model Reduction: Balanced truncation and projection methods
  • Numerical Analysis: Matrix exponentials and matrix functions
  • Optimal Control: Riccati equations and LQR problems
Special Cases and Variants
  • Lyapunov Equation: AX + XAT = C (when B = -AT)
  • Stein Equation: AXB - X = C (discrete-time version)
  • Generalized Sylvester: AXB + CXD = E
  • Coupled Sylvester: System of multiple Sylvester equations
Computational Complexity
  • Direct Method: O(m³n³) using Kronecker products
  • Schur Method: O(m³ + n³ + m²n) operations
  • Iterative Methods: O(kmn) per iteration, where k depends on convergence
  • Memory Requirements: O(mn) for solution, O(m² + n²) for decompositions

Mathematical Disclaimer: This calculator provides numerical solutions to the Sylvester equation for educational and research purposes. Results are subject to floating-point precision limitations. For large matrices or ill-conditioned problems, specialized numerical software may be required. Always verify critical results independently.

Sylvester Equation: \( AX + XB = C \)

Vectorized Form: \( (I \otimes A + B^T \otimes I) \cdot \text{vec}(X) = \text{vec}(C) \)

Solvability Condition: For a unique solution, eigenvalues of A and -B must not sum to zero: \( \lambda_i(A) + \lambda_j(-B) \ne 0 \) for all i, j

What Is the Sylvester Equation Calculator?

The Sylvester Equation Calculator helps solve equations of the form \( AX + XB = C \), where A, B, and C are known matrices and X is the unknown matrix to be found. This type of equation is widely used in control systems, signal processing, and numerical Linear Algebra.

This tool provides a guided way to input matrices, select solution methods, and receive detailed output including step-by-step explanations, verification, and matrix properties. It is especially useful for students, engineers, and researchers working with advanced matrix equations.

How to Use the Calculator

Follow these steps to find the solution matrix X:

  • Enter the dimensions of matrices A and B using the provided fields.
  • Click Generate Matrix Input Fields to create entry boxes for matrices A, B, and C.
  • Input the matrix values manually or click Load Example to autofill with sample data.
  • Choose a solution method: Direct (Kronecker Product), Iterative, or Schur Decomposition.
  • Set the desired decimal precision and optional checkboxes for steps and verification.
  • Click Solve Sylvester Equation to compute the solution.

What You’ll Get

After solving, the calculator displays:

  • Solution Status: Indicates whether a unique solution was found.
  • Solution Matrix X: The matrix that satisfies \( AX + XB = C \).
  • Solution Steps: A clear breakdown of the method used.
  • Verification: Confirms the solution by substituting it back into the original equation.
  • Eigenvalue Analysis: Shows whether A and -B meet the solvability condition.
  • Matrix Properties: Includes determinants and Frobenius norms of the matrices involved.

Why This Calculator Is Useful

Solving the Sylvester equation by hand can be time-consuming and prone to error. This tool automates the process and ensures accuracy. It's particularly helpful when working with matrix methods like:

  • LU matrix factorization – used in solving linear systems and computing matrix inverses
  • Matrix inverse tool – when manipulating the equation to isolate X
  • Gauss-Jordan method – as part of solving the vectorized form of the equation
  • Matrix eigenvalue analysis – to check the solvability condition

The calculator also connects with broader concepts such as matrix decomposition, matrix exponentiation, and linear system solvers, making it a valuable companion to Other tools like the Diagonalize Matrix Calculator, QR Factorization Tool, and Matrix Rank Finder.

Frequently Asked Questions (FAQ)

  • What is the Sylvester equation used for?
    It appears in control theory, signal processing, and model reduction where matrix equations need to be solved efficiently.
  • What conditions must be met for a unique solution?
    The matrices A and -B must not share any eigenvalues. If they do, the equation may not have a unique solution.
  • Which method should I choose?
    The Direct method is best for small matrices. Use the Iterative option for experimentation or the Schur method for a theoretical alternative (mapped to the direct method in this tool).
  • Can I verify my solution?
    Yes, the tool checks if the computed X satisfies the equation by computing \( AX + XB \) and comparing it to C.
  • What if I get an error?
    Check your matrix dimensions and ensure that all fields are filled with valid numbers. Make sure the solvability condition is met.

Conclusion

The Sylvester Equation Calculator is a fast and effective way to solve matrix equations of the form \( AX + XB = C \). With features like multiple solution methods, step-by-step guidance, and solution verification, it makes advanced matrix algebra more accessible and practical. Whether you’re exploring matrix eigenvalue processes or applying LU decomposition steps, this calculator supports your work with reliable results.