2 by 2 Matrix Eigenvalue Calculator
Calculate real or complex-conjugate eigenvalues and discriminant of a real 2×2 matrix.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Use the characteristic polynomial to separate real and imaginary components without forcing complex roots to be real.
Eigenvalues=trace/2 ± sqrt(((a−d)/2)²+bc)
Worked example
Matrix2,1;1,2 has eigenvalues3 and1; matrix0,−1;1,0 has eigenvalues±i.
Assumptions and input details
Finite real entries of magnitude≤1e12. Eigenvectors, Jordan form and numerical balancing are excluded. A repeated root does not imply two independent eigenvectors.
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.
Add, subtract, multiply or divide two complex numbers. Enter real and imaginary parts separately, then inspect magnitude, argument and an equal-scale complex-plane diagram.
Score a two-dimensional observation against two Gaussian classes with a common positive-definite covariance.
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