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Enter your values, then click Calculate result.How this calculator helps
Find eigenvalues of a numeric 2×2 real matrix, with real unit eigenvectors or a labeled complex conjugate pair and trace checks. Enter four real numeric entries in row order: upper left, upper right, lower left and lower right. The calculation is restricted to a two-by-two matrix. It does not accept a pasted arbitrary-size array, symbolic variables or expressions. Entries must be finite and between −10⁶ and 10⁶. Confirm row placement before interpreting the result because transposition can change eigenvectors even when eigenvalues stay unchanged.
How to use it
- 1
Choose the correct input basis for eigenvalue calculator — 2×2 matrices and enter the values described below. The demonstration defaults are examples, not independently verified personal measurements.
- 2
Review characteristic equation before submitting. Match the selected units and roles to the original source record, including any signs or percentage conventions.
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Click Calculate result to submit the current inputs. Editing a field preserves the previous submitted output until you calculate again; the status message identifies that pending change.
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Compare the labeled result with the worked example and independent verification checks. Review numerical interpretation before copying it into another worksheet.
Formula and methodology
For A = [[a,b],[c,d]], trace t = a+d and determinant Δ = ad−bc. Eigenvalues solve λ²−tλ+Δ = 0, with discriminant (a−d)²+4bc.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
The diagonal matrix [[2,0],[0,3]] has eigenvalues 3 and 2. Unit eigenvectors can be (0,1) and (−1,0), respectively; changing a vector’s sign preserves its eigenvector property. The sum of eigenvalues is 5, matching trace, and their product is 6, matching determinant.
How to interpret your result
For each real value, the worksheet constructs a nonzero vector in the null space of A−λI and normalizes its Euclidean length to one. Multiplying an eigenvector by any nonzero scalar gives another valid eigenvector, so signs and scale can differ from a textbook answer. Check the direction through Av = λv instead of expecting one identical written vector.
For different inputs or formulas, use Matrix Multiplication Calculator; Inverse Matrix Calculator.
Related questions this calculator covers
- eigenvalue calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Diagonal | [[2,0],[0,3]] gives 2 and 3. |
| Rotation | [[0,−1],[1,0]] gives ±i. |
| Scalar | [[4,0],[0,4]] gives repeated 4 and unrestricted nonzero eigenvectors. |
Common mistakes to avoid
- Reordering matrix entries while entering them.
- Rejecting a valid eigenvector because its sign differs.
- Assuming a repeated value proves diagonalizability.
Verify the sum against trace and product against determinant. For each displayed real vector, multiply the original matrix by that vector and compare with eigenvalue times vector. A small residual is a numerical check, not a symbolic proof. Test a diagonal, scalar and complex-pair example to distinguish the three principal result branches.
Authoritative reference. Method references reviewed on 5 October 2026. SolvePilot supplies the original examples and bounded browser implementation. Review by Mohammad Qasim covers editorial scope and arithmetic, not individual professional approval. The cited reference provides method or unit context rather than certifying the entered measurements or assumptions.What can affect the result?
Matrix entry order
Enter four real numeric entries in row order: upper left, upper right, lower left and lower right. The calculation is restricted to a two-by-two matrix. It does not accept a pasted arbitrary-size array, symbolic variables or expressions. Entries must be finite and between −10⁶ and 10⁶. Confirm row placement before interpreting the result because transposition can change eigenvectors even when eigenvalues stay unchanged.
Characteristic equation
An eigenvalue makes A−λI singular. For a two-by-two matrix, expanding its determinant produces the displayed quadratic equation. The trace and determinant therefore determine the eigenvalue sum and product. A positive discriminant yields two real values, zero gives a repeated value, and a negative discriminant yields a complex conjugate pair. The page labels that pair rather than discarding an imaginary component.
Real eigenvectors
For each real value, the worksheet constructs a nonzero vector in the null space of A−λI and normalizes its Euclidean length to one. Multiplying an eigenvector by any nonzero scalar gives another valid eigenvector, so signs and scale can differ from a textbook answer. Check the direction through Av = λv instead of expecting one identical written vector.
Repeated and scalar cases
A repeated eigenvalue need not have two independent eigenvectors. A defective matrix can return the same eigenspace for both reported values. A scalar multiple of the identity has every nonzero vector as an eigenvector, which is stated explicitly instead of pretending there is one distinguished direction. The worksheet does not construct a Jordan form or certify diagonalizability for an arbitrary matrix.
Numerical interpretation
Near repeated values, tiny entry changes can substantially change eigenvectors. Ordinary floating-point calculations also lose relative precision in badly conditioned examples. This tool is an educational two-by-two worksheet, not an arbitrary-precision linear-algebra package. Complex eigenvectors are not constructed. Retain the original matrix and inspect residuals when numerical accuracy matters rather than treating eight displayed decimals as an error bound.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Near repeated values, tiny entry changes can substantially change eigenvectors. Ordinary floating-point calculations also lose relative precision in badly conditioned examples. This tool is an educational two-by-two worksheet, not an arbitrary-precision linear-algebra package. Complex eigenvectors are not constructed. Retain the original matrix and inspect residuals when numerical accuracy matters rather than treating eight displayed decimals as an error bound.
Sources and review information
Frequently asked questions
Can I enter a 3×3 matrix?＋
No. This calculator deliberately handles numeric two-by-two real matrices only.
Why are my vector signs different?＋
Eigenvectors are unchanged as directions when multiplied by a nonzero scalar, including minus one.
What does an imaginary eigenvalue mean?＋
The characteristic quadratic has negative discriminant. The page reports a conjugate pair but does not build complex eigenvectors.
Does a repeated value mean two independent vectors?＋
No. A repeated eigenvalue may have only one independent eigendirection.
What happens for the identity matrix?＋
Both eigenvalues are one and every nonzero vector is an eigenvector, so the worksheet reports that general case.