Enter a matrix and candidate eigenvector. Get eigenvalue checks, residuals, and worked steps fast clearly. Built for physics study, verification, and quick classroom practice.
| Case | Matrix A | Eigenvector v | Av | Eigenvalue λ |
|---|---|---|---|---|
| Example 1 | [[4,1,0],[0,4,0],[0,0,2]] | [1,0,0] | [4,0,0] | 4 |
| Example 2 | [[3,0],[0,5]] | [0,1] | [0,5] | 5 |
| Example 3 | [[2,0,0],[0,2,0],[0,0,2]] | [2,-1,4] | [4,-2,8] | 2 |
The calculator uses the eigenvector relation Av = λv.
First, it multiplies the matrix A by the candidate vector v.
Next, it computes each available ratio λ = (Av)i / vi when vi ≠ 0.
If all valid ratios match within the chosen tolerance, the common value is the eigenvalue.
If a vector component is zero, then the matching component of Av must also be zero.
The residual check is r = Av - λv.
The residual norm ‖r‖ shows how closely the candidate pair satisfies the eigenvalue equation.
Eigenvalues appear everywhere in physics. They describe measurable quantities and natural system behavior. This calculator helps you extract an eigenvalue from a known eigenvector. It works by testing the matrix equation directly. The method is clear and fast. It also shows whether your chosen vector truly behaves like an eigenvector.
The input matrix can represent a linear operator, transition rule, or physical model. The input vector is your candidate eigenvector. The calculator multiplies the matrix by the vector. Then it compares the result with the original vector. If both point in the same scaled direction, the scale factor is the eigenvalue. That is the core idea behind the eigenvalue equation.
In quantum mechanics, eigenvalues often represent measurable quantities such as energy. In vibration analysis, they relate to natural frequencies and mode shapes. In rigid body motion, they help describe principal directions. In wave problems, they help identify stable patterns and resonant behavior. Because of these links, checking an eigenpair quickly can save time during homework, lab analysis, and model verification.
Real data is often noisy. Numerical models also introduce rounding error. That is why the residual norm is valuable. A very small residual means the matrix equation is nearly satisfied. A large residual means the candidate vector is not an eigenvector for the matrix, or the data contains mistakes. This extra check makes the result more reliable.
This calculator is useful for students, teachers, and engineers. It reduces manual ratio checking. It also highlights zero-component issues, which are often missed in hand calculations. The component table gives a transparent audit trail. You can inspect each ratio, compare values, and export the calculation for notes or reports. That makes the tool practical for classroom work and technical review.
It finds the eigenvalue linked to a supplied matrix and candidate eigenvector. It also checks whether the pair satisfies the eigenvalue equation within your selected tolerance.
No. The zero vector is never an eigenvector. An eigenvector must be nonzero, or the eigenvalue relation loses meaning.
Each nonzero vector component gives a ratio (Av)i / vi. If the vector is a true eigenvector, those ratios should agree.
If vi is zero, then (Av)i must also be zero. Otherwise, the candidate vector fails the eigenvector test for that matrix.
The residual norm measures the difference between Av and λv. A smaller value means the pair matches the eigenvalue equation more closely.
Yes. Eigenvalues appear in quantum mechanics, vibration analysis, normal modes, stability studies, and many operator-based physical models.
Yes. That is why the tolerance field is included. It lets you judge near matches when measurements or numerical approximations are involved.
A failed check means the candidate vector is not an exact eigenvector for the entered matrix, or the inputs contain an error.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.