Eigenvalue Calculator
Calculate the eigenvalues of a square matrix used in dimensionality reduction.
About Eigenvalue Calculator
Eigenvalues of a 2×2 matrix are found by solving the characteristic equation det(A − λI) = 0, which reduces to λ² − trace×λ + det = 0. Using the quadratic formula: λ = (trace ± √(trace² − 4×det)) / 2. Eigenvalues are fundamental in PCA, spectral clustering, and stability analysis of dynamical systems.
Formula
Characteristic Equation
lambda = (trace ± sqrt(trace^2 - 4*det)) / 2
- trace
- Sum of diagonal elements (a + d)
- det
- Determinant of the matrix
Worked example
With the default values below, Eigenvalue Calculator returns Eigenvalue λ₁ of 5.000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (row1,col1)
- 4
- b (row1,col2)
- 1
- c (row2,col1)
- 2
- d (row2,col2)
- 3
Results
- Eigenvalue λ₁
- 5.000000
- Eigenvalue λ₂
- 2.000000
- Trace
- 7.000000
- Determinant
- 10.000000
Step by step
- 1Trace (a + d)= 7.000000
- 2Determinant (ad − bc)= 10.000000
- 3Discriminant: trace² − 4×det= 9.000000
- 4λ₁ = (trace + √discriminant) / 2= 5.000000
- 5λ₂ = (trace − √discriminant) / 2= 2.000000
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How are eigenvalues used in PCA?
In PCA, eigenvalues of the covariance matrix represent the variance explained by each principal component. The largest eigenvalues correspond to the directions of maximum variance in the data.
What if the discriminant is negative?
A negative discriminant means the eigenvalues are complex conjugates. This occurs in matrices representing rotations or oscillatory systems and indicates no real principal directions exist.