Eigenvalue Calculator
Calculate eigenvalues and eigenvectors for 2x2 matrices step-by-step using characteristic equation det(A - λI) = 0.
2×2 Eigenvalue & Eigenvector Engine
Enter elements for 2x2 matrix A.
Eigenvalue Output
Quick Summary
Our eigenvalue calculator 2x2 matrix characteristic polynomial steps tool calculates the eigenvalues $\lambdA1, \lambdA2$ and eigenvectors $\mathbf{v}_1, \mathbf{v}_2$ of $2 \times 2$ matrices. It solves the characteristic polynomial $\det(A - \lambda I) = 0 \iff \lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$, verifying matrix trace and determinant relationships.
How It Works: Characteristic Equation & Eigenvectors
Eigenvalues represent stretch factors along directions (eigenvectors) invariant under matrix transformation:
1. **Fundamental Equation:** $A \mathbf{v} = \lambda \mathbf{v} \iff (A - \lambda I) \mathbf{v} = \mathbf{0}$.
2. **Characteristic Polynomial:** Set determinant equal to zero: $\det(A - \lambda I) = 0 \implies \lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$.
3. **Matrix Trace & Determinant Properties:** Sum of eigenvalues equals trace $\lambdA1 + \lambdA2 = \text{tr}(A) = a_{11} + a_{22}$; product equals determinant $\lambdA1 \cdot \lambdA2 = \det(A)$.
4. **Eigenvalues Solution:** Use quadratic formula $\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}$.
5. **Eigenvector Null-Space Computation:** Solve non-trivial null space kernel for each $\lambda_i$: $(A - \lambda_i I)\mathbf{v}_i = \mathbf{0}$.
Formula Explanation
Your eigenvalue calculations follow quadratic characteristic formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for finding eigenvalues of 2×2 matrix $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$:
- Step 1 (Calculate Matrix Trace): $\text{tr}(A) = 4 + 3 = \mathbf{7}$.
- Step 2 (Calculate Matrix Determinant): $\det(A) = (4)(3) - (1)(2) = 12 - 2 = \mathbf{10}$.
- Step 3 (Form Characteristic Polynomial): $\lambda^2 - 7\lambda + 10 = 0$.
- Step 4 (Solve Quadratic for Eigenvalues): $(\lambda - 5)(\lambda - 2) = 0 \implies \mathbf{\lambdA1 = 5, \lambdA2 = 2}$.
- Step 5 (Compute Eigenvectors): For $\lambdA1 = 5$: $\begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \mathbf{v}_1 = \mathbf{0} \implies \mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$. For $\lambdA2 = 2$: $\mathbf{v}_2 = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$, delivering **λ₁ = 5, λ2 = 2**!
Calculation Examples: Real-World Scenario Comparison
Compare matrix inputs, characteristic polynomials, traces, determinants, and calculated eigenvalues:
| Matrix A | tr(A) | det(A) | Characteristic Equation | Eigenvalues (λ&sb1;, λ&sb2;) |
|---|---|---|---|---|
| [[4, 1], [2, 3]] | 7 | 10 | λ² - 7λ + 10 = 0 | λ&sb1; = 5, λ&sb2; = 2 |
| [[1, 0], [0, 1]] | 2 | 1 | λ² - 2λ + 1 = 0 | λ = 1 (Double Root) |
| [[0, -1], [1, 0]] | 0 | 1 | λ² + 1 = 0 | λ = ± i (Complex Pair) |
| [[3, 5], [0, 7]] | 10 | 21 | λ² - 10λ + 21 = 0 | λ&sb1; = 7, λ&sb2; = 3 |
Benefits of Using the Eigenvalue Calculator
Utilizing this calculator provides essential PCA dimensionality reduction, Google PageRank, and vibration analysis advantages:
- Principal Component Analysis (PCA): Identifies dominant variance axes in machine learning feature extraction.
- Vibration & Structural Stability Analysis: Computes natural resonance frequencies ($\omega = \sqrt{\lambda}$) in mechanical systems.
- Google PageRank Algorithm: Computes stationary state dominant eigenvector ($\lambda = 1$) for web page ranking.
- Quantum Mechanics Wavefunctions: Solves Schrödinger energy eigenvalue equations ($\hat{H} \psi = E \psi$).
Frequently Asked Questions (FAQ)
What are eigenvalues and eigenvectors?
An eigenvector v is a non-zero vector that changes only by a scalar factor λ (eigenvalue) when a linear transformation A is applied: A v = λ v.
What is the characteristic equation of a matrix?
The characteristic equation det(A - λI) = 0 is a polynomial equation whose roots are the eigenvalues of matrix A.
What is matrix trace tr(A)?
The trace of a square matrix is the sum of its main diagonal elements (tr(A) = a11 + a22). The sum of all eigenvalues equals tr(A).
How are eigenvalues related to determinant det(A)?
The product of all eigenvalues equals the matrix determinant: λ1 × λ2 = det(A).
Can eigenvalues be complex numbers?
YES! If the characteristic equation discriminant b^2 - 4ac < 0, the eigenvalues appear as complex conjugate pairs (e.g. rotation matrices).
What are eigenvalues of a diagonal or triangular matrix?
For diagonal or upper/lower triangular matrices, the eigenvalues are simply the entries on the main diagonal.
What are eigenvalues of a symmetric matrix?
All eigenvalues of a real symmetric matrix (A^T = A) are guaranteed to be REAL numbers.
What does an eigenvalue of 0 mean?
If λ = 0 is an eigenvalue, the matrix is singular (det(A) = 0) and non-invertible.
How is PCA (Principal Component Analysis) related to eigenvalues?
PCA finds the eigenvectors and eigenvalues of the data covariance matrix; eigenvectors give principal directions and eigenvalues give variance magnitude.
How do you find eigenvectors for a given eigenvalue λ?
Substitute λ back into (A - λI) v = 0 and solve the linear homogeneous system for vector v.