Welcome to the Hadamard Ratio Calculator. This tool is designed to help you quickly determine the Hadamard ratio of a given matrix. The Hadamard ratio is a measure that compares the determinant of a matrix to the product of the lengths (norms) of its column vectors. It is especially useful in linear algebra, numerical analysis, and optimization, providing insight into how “orthogonal” or “well-conditioned” a matrix is.
The calculator is simple to use: enter your matrix, and it will instantly compute the ratio. You can jump straight to the calculation, or keep reading to learn more about the formula, the meaning of its components, and a worked example.
Understanding the Formula
The formula for the Hadamard Ratio is:
H = det(A) / (Π ||vᵢ||)
Variables:
- H: Hadamard Ratio
- det(A): Determinant of the matrix A
- Π: Product operator, representing multiplication of terms
- ||vᵢ||: Euclidean norm (length) of each column vector vᵢ of the matrix A
In simple terms, this formula checks how close a matrix is to being perfectly orthogonal. If the columns of a matrix are mutually orthogonal, the Hadamard ratio reaches its maximum possible value (close to 1). When the columns are nearly dependent or aligned, the ratio decreases, highlighting potential numerical instability.
Parameters Explained
det(A) (Determinant of the matrix)
A scalar value that summarizes properties of the matrix, such as invertibility. If det(A) = 0, the matrix is singular and the Hadamard ratio will be 0.
||vᵢ|| (Euclidean norm of a column vector)
This measures the length of each column vector. It is calculated as the square root of the sum of squares of the vector’s components.
Π (Product operator)
This simply means multiplying all the column vector norms together. For example, with three column vectors v₁, v₂, and v₃, you multiply ||v₁|| × ||v₂|| × ||v₃||.
H (Hadamard Ratio)
The final result. A value close to 1 suggests that the matrix has well-spaced, nearly orthogonal columns. Smaller values indicate higher dependency among columns.
How to Use the Hadamard Ratio Calculator — Step-by-Step Example
Let’s calculate the Hadamard ratio for a 2×2 matrix:
A =
[ 1 2 ]
[ 3 4 ]
Step 1: Find the determinant
det(A) = (1×4) − (2×3) = 4 − 6 = −2
Step 2: Find the norms of each column
v₁ = [1, 3] → ||v₁|| = √(1² + 3²) = √10 ≈ 3.162
v₂ = [2, 4] → ||v₂|| = √(2² + 4²) = √20 ≈ 4.472
Step 3: Multiply the norms
Π ||vᵢ|| = 3.162 × 4.472 ≈ 14.142
Step 4: Apply the formula
H = det(A) / (Π ||vᵢ||) = (−2) / (14.142) ≈ −0.141
Final Result:
The Hadamard ratio is approximately −0.141. Since the ratio is far from 1, this matrix has highly dependent columns.
Additional Information
Some useful points about the Hadamard Ratio:
| Range of H | Interpretation |
|---|---|
| Close to 1 | Columns are nearly orthogonal (stable matrix) |
| Around 0 | High dependency among columns (unstable matrix) |
| Negative | Possible orientation issues, but magnitude still indicates dependency |
This ratio is commonly used in numerical optimization, matrix condition checks, and to evaluate the geometry of lattice structures.
FAQs
It indicates how “independent” the columns of a matrix are. A higher ratio means stronger independence, while a lower ratio suggests dependency.
Yes, because the determinant can be negative. However, the magnitude of H is often more important than its sign when analyzing column dependence.
It is widely used in numerical analysis, lattice theory, and linear algebra to study stability, orthogonality, and optimization problems.