Hadamard Ratio Calculator

Formula

To calculate the Hadamard Ratio (\(H\)):

\[ H = \frac{D}{P} \]

Where:

\(H\) is the Hadamard Ratio
\(D\) is the determinant of the matrix
\(P\) is the product of Euclidean norms of the columns of the matrix

What is the Hadamard Ratio?

The Hadamard Ratio is a measure used in linear algebra to evaluate the orthogonality of a matrix. It is defined as the ratio of the determinant of the matrix to the product of the Euclidean norms of its columns. A Hadamard Ratio close to 1 indicates that the matrix is nearly orthogonal, meaning its columns are nearly perpendicular to each other. This ratio is particularly useful in numerical analysis and optimization problems where the condition number of a matrix plays a crucial role.

Example Calculation 1

Let's assume the following values:

Determinant (\(D\)) = 10
Product of Euclidean Norms (\(P\)) = 20

Using the formula:

\[ H = \frac{10}{20} = 0.5 \]

The Hadamard Ratio is 0.5.

Example Calculation 2

Let's assume the following values:

Determinant (\(D\)) = 15
Product of Euclidean Norms (\(P\)) = 25

Using the formula:

\[ H = \frac{15}{25} = 0.6 \]

The Hadamard Ratio is 0.6.