Hadamard Ratio Calculator
Calculate the Hadamard ratio to measure matrix column orthogonality.
Matrix Input
Example Matrices
Calculation Results
Enter a square matrix and click "Calculate Hadamard Ratio" to see results
Hadamard Ratio Calculator
The Hadamard ratio is a numerical value that shows how close a square matrix is to having perfectly orthogonal columns. A ratio of 1 means the matrix has ideal orthogonality. Lower values indicate increasing correlation between columns, which can affect numerical stability in computations.
This calculator computes the Hadamard ratio quickly without requiring manual determinant or norm calculations. It's particularly useful in linear algebra, numerical analysis, signal processing, and optimization problems where matrix conditioning matters.
How the Calculator Works
Hadamard Ratio Formula
The calculator uses the fundamental Hadamard inequality formula:
H(A) = |det(A)| / ∏||aᵢ||
Where:
det(A)is the determinant of matrix A||aᵢ||is the Euclidean norm of column i∏represents the product of all column norms
The result is always between 0 and 1, with 1 indicating perfect orthogonality.
Example Calculations
Sample Matrix (3×3)
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | 1 | 0 | 1 |
| Row 2 | 0 | 1 | 1 |
| Row 3 | 1 | 1 | 0 |
Intermediate Calculations
| Determinant |det(A)| | 2.0000 |
| Column 1 Norm ||a₁|| | 1.4142 |
| Column 2 Norm ||a₂|| | 1.4142 |
| Column 3 Norm ||a₃|| | 1.4142 |
| Product of Norms | 2.8284 |
Final Result
This matrix has moderate orthogonality. An identity matrix would give H(A) = 1.0 (perfect orthogonality), while a singular matrix would give H(A) = 0.