Compare right and left eigenvectors with clarity. Inspect null spaces, transpose behavior, and repeated roots. Use this calculator for practice, validation, and deeper understanding.
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| 4 | 1 |
| 2 | 3 |
Matrix Size: 2x2
Trace: 7
Determinant: 10
Symmetric: No
Characteristic Polynomial: λ² - (7)λ + (10) = 0
The matrix has two distinct real eigenvalues.
Right eigenvectors solve A v = λ v.
Left eigenvectors solve wT A = λ wT.
They are right eigenvectors of AT.
| Eigenvalue | Algebraic Multiplicity | Right Geometric Multiplicity | Left Geometric Multiplicity | Right Eigenvector Basis | Left Eigenvector Basis | Left·Right |
|---|---|---|---|---|---|---|
| 2 | 1 | 1 | 1 | [-0.447214, 0.894427] | [-0.707107, 0.707107] | 0.948683 |
| 5 | 1 | 1 | 1 | [0.707107, 0.707107] | [0.894427, 0.447214] | 0.948683 |
| Case | Matrix | Main Idea |
|---|---|---|
| Example 1 | [[4, 1], [2, 3]] | Two distinct real eigenvalues. Left and right vectors differ. |
| Example 2 | [[2, 1], [0, 2]] | Repeated eigenvalue. Useful for defective matrix checks. |
| Example 3 | [[3, 1, 0], [0, 2, 1], [0, 0, 1]] | Upper triangular matrix. Eigenvalues match the diagonal. |
Right eigenvectors satisfy A v = λ v.
That becomes (A - λI) v = 0.
So the right eigenvector basis comes from the null space of A - λI.
Left eigenvectors satisfy wTA = λ wT.
This is equivalent to (AT - λI) w = 0.
So the left eigenvector basis comes from the null space of AT - λI.
For 2×2 matrices, eigenvalues come from det(A - λI) = 0.
For 3×3 matrices, the calculator builds the cubic characteristic polynomial and solves real roots numerically.
Geometric multiplicity equals the number of null space basis vectors.
Algebraic multiplicity equals the repeated count of the eigenvalue in the characteristic polynomial.
Right and left eigenvectors are not always the same. They match for many symmetric matrices. They differ for many non symmetric matrices. That difference matters in applied linear algebra. It appears in control, Markov models, vibration work, and matrix perturbation analysis.
Right eigenvectors describe directions that stay aligned after matrix action. The matrix only stretches or flips those directions by the eigenvalue. They are the standard vectors students first study. They support diagonalization, powers of matrices, and stability checks.
Left eigenvectors come from the transpose system. They act like row based partners to right eigenvectors. In many advanced problems, they give sensitivity information. They also help define biorthogonal bases for non normal matrices. That makes them important for deeper interpretation.
This right vs left eigenvectors calculator first builds the characteristic polynomial. Next it finds real eigenvalues. Then it forms the matrices A minus lambda I and A transpose minus lambda I. After that, it computes null spaces with row reduction. Those null spaces produce the right and left eigenvector bases.
Check algebraic multiplicity and geometric multiplicity together. If they differ, the matrix may be defective. Compare the left basis and right basis. If the matrix is symmetric, the two bases usually agree up to scaling. If not, expect differences. That contrast is the main teaching value here.
Use it for homework checking, class demonstrations, and quick numeric validation. It is also useful when learning transpose relationships, repeated roots, and defective behavior. The export tools help you save results for reports, assignments, and revision sheets.
A right eigenvector is a nonzero vector v that satisfies A v = λ v. The matrix changes its magnitude or sign, but not its direction.
A left eigenvector is a nonzero vector w that satisfies wTA = λwT. It is equivalent to a right eigenvector of the transpose matrix.
They differ when a matrix is not symmetric. The null space of A - λI can differ from the null space of AT - λI.
Yes. It supports both 2×2 and 3×3 real matrices. It displays real eigenvalues and their real left and right eigenvector bases.
The calculator focuses on real outputs. If some eigenvalues are complex, the page notes that they are omitted from the displayed real basis results.
Geometric multiplicity is the dimension of the eigenspace. In this calculator, it equals the number of basis vectors found in the null space.
Algebraic multiplicity is how many times an eigenvalue repeats in the characteristic polynomial. It helps identify repeated roots and defective cases.
They usually match up to scaling for symmetric matrices. In that case, A and AT have the same eigenspaces for each eigenvalue.
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.