Sylvester Matrix Equation Calculator
Solve AX+XB=C for a real 2×2 unknown matrix with a unique well-conditioned solution.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Build coefficients directly for row-major unknowns and use pivoted elimination with a residual readback.
Vectorize AX+XB=C into four simultaneous linear equations
Worked example
Diagonal A(1,2), B(3,4) and C(4,5;5,6) give all entries of X equal1.
Assumptions and input details
Real 2×2 matrices, unique solution and pivot conditioning threshold. Singular or ill-conditioned coefficient systems are rejected instead of returning an arbitrary solution.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Add, subtract or multiply two 2×2 matrices, or find the inverse of A. Enter each cell separately and follow row-by-column arithmetic in the result table.
Solve two linear equations in x and y. See determinants, substitution checks and the two lines, including parallel or coincident cases.
Calculate real or complex-conjugate eigenvalues and discriminant of a real 2×2 matrix.
Reserved advertising space.