Calculator Inputs
Example Data Table
Reference values for several common Hilbert matrix orders.
| Order n | Determinant | log10(det) | First Row Preview |
|---|---|---|---|
| 2 | 8.33333333 × 10-2 | -1.07918125 | 1.000000, 0.500000 |
| 3 | 4.62962963 × 10-4 | -3.33445375 | 1.000000, 0.500000, 0.333333 |
| 4 | 1.65343915 × 10-7 | -6.78161178 | 1.000000, 0.500000, 0.333333, 0.250000 |
| 5 | 3.74929513 × 10-12 | -11.42605037 | 1.000000, 0.500000, 0.333333, 0.250000, 0.200000 |
| 6 | 5.36729989 × 10-18 | -17.27024414 | 1.000000, 0.500000, 0.333333, 0.250000, 0.200000, 0.166667 |
Formula Used
The Hilbert matrix of order n has entries H(i,j) = 1 / (i + j - 1). Its determinant has a closed-form product formula that avoids expensive symbolic elimination for many practical calculations.
Determinant formula:
det(H_n) = [∏ from k=1 to n-1 (k!)^4] / [∏ from k=1 to 2n-1 k!]
Because Hilbert determinants become extremely small, this calculator also evaluates the base-10 logarithm:
log10(det(H_n)) = 4·Σ log10(k!) − Σ log10(k!), using the correct numerator range 1..n−1 and denominator range 1..2n−1.
This log-domain approach reduces underflow and provides stable display values for larger matrix orders.
How to Use This Calculator
- Enter the Hilbert matrix order n.
- Choose how many rows and columns to preview.
- Select the number of digits for displayed values.
- Optionally enable the reciprocal result for condition comparisons.
- Press Calculate Determinant to show results above the form.
- Review the determinant, log value, and matrix preview.
- Use the CSV or PDF buttons to export results or example data.
Why Hilbert Determinants Matter
Hilbert matrices are classic examples of ill-conditioned matrices in numerical linear algebra. Their determinants shrink rapidly as order increases, making them useful for testing algorithms, precision handling, and symbolic identities.
This calculator supports exact-fraction display for smaller orders and scientific notation with logarithmic scaling for larger ones. That combination helps students, teachers, and researchers inspect both structure and magnitude quickly.
Frequently Asked Questions
1. What is a Hilbert matrix?
A Hilbert matrix is a square matrix whose entries follow 1/(i + j − 1). It is symmetric, positive definite, and famously ill-conditioned.
2. Why is the determinant so small?
As the order grows, the matrix columns become nearly dependent numerically. That structure causes the determinant to decrease very quickly.
3. Why show a logarithm of the determinant?
Very small determinants can underflow in ordinary decimal display. A base-10 logarithm preserves scale information clearly and remains stable for larger orders.
4. Is the determinant always positive?
Yes. Every finite Hilbert matrix is positive definite, so all of its determinants are positive.
5. Does this tool compute the matrix directly?
It generates a matrix preview directly, but the determinant uses the closed-form product formula. That method is faster and usually more stable.
6. When is the exact fraction shown?
The exact fraction is shown for smaller matrix orders where standard floating calculations remain manageable. Larger orders are presented with scientific notation instead.
7. What does the reciprocal help with?
The reciprocal highlights how tiny the determinant is. It can be useful when comparing growth rates or discussing conditioning behavior.
8. Can I use this for teaching and benchmarking?
Yes. Hilbert matrices are standard examples in numerical analysis, so this calculator works well for demonstrations, assignments, and algorithm stress tests.