Calculation Result
Matrix Size
Largest Eigenvalue
Trace
| Component | Eigenvalue | Variance Share | Cumulative Share |
|---|
Calculator Input
Enter a symmetric correlation matrix. Diagonal values should be 1. Off-diagonal values usually stay between -1 and 1.
Example Data Table
This example shows a valid 3 × 3 correlation matrix.
| Variable A | Variable B | Variable C | |
|---|---|---|---|
| Variable A | 1.00 | 0.72 | 0.48 |
| Variable B | 0.72 | 1.00 | 0.51 |
| Variable C | 0.48 | 0.51 | 1.00 |
Formula Used
For a correlation matrix R, eigenvalues come from the characteristic equation:
det(R - λI) = 0
Here, λ represents each eigenvalue, and I is the identity matrix. The sum of all eigenvalues equals the trace of the correlation matrix. For a valid correlation matrix, that trace equals the number of variables.
Variance share for each component is:
Variance Share = λ / Σλ
Cumulative variance is the running total of those shares after sorting eigenvalues from largest to smallest.
How to Use This Calculator
- Select the matrix size you need.
- Enter diagonal values as 1.
- Fill the off-diagonal correlation values.
- Keep matching mirrored cells equal.
- Load a preset if you want a quick example.
- Click Calculate Eigenvalues.
- Review eigenvalues, variance shares, and the chart.
- Export the output as CSV or PDF.
8 FAQs
1. What does an eigenvalue represent here?
An eigenvalue shows how much variance a principal direction explains in the correlation matrix. Larger values indicate stronger shared structure among variables.
2. Why must the matrix be symmetric?
Correlation matrices are symmetric by definition. The correlation between A and B must equal the correlation between B and A.
3. Why are diagonal entries usually 1?
Each variable is perfectly correlated with itself. That makes every diagonal value equal to one in a standard correlation matrix.
4. What does the trace tell me?
The trace is the sum of diagonal values. In a correlation matrix, it equals the number of variables, and it also equals the sum of eigenvalues.
5. Why sort eigenvalues from largest to smallest?
Sorting helps you see which latent directions explain the most variance first. This is standard for principal component style interpretation.
6. Can eigenvalues ever be negative?
A valid correlation matrix should be positive semidefinite, so eigenvalues should not be negative. Small negatives may appear from rounding or invalid inputs.
7. What does a dominant first eigenvalue mean?
It suggests one main latent factor explains much of the shared variance. This often indicates strong common structure across variables.
8. When is this calculator useful?
It helps in statistics, factor exploration, PCA preparation, teaching, and quick checks of whether variables cluster around strong shared patterns.