Cauchy Matrix Determinant

Worked Example

Consider a small 2 × 2 example with x = (1, 2) and y = (0, 1). The Cauchy matrix C is constructed as follows:

C_11 = 1 / (x_1 + y_1) = 1 / (1 + 0) = 1 C_12 = 1 / (x_1 + y_2) = 1 / (1 + 1) = 1/2 C_21 = 1 / (x_2 + y_1) = 1 / (2 + 0) = 1/2 C_22 = 1 / (x_2 + y_2) = 1 / (2 + 1) = 1/3 C = [[ 1,   1/2 ], [ 1/2, 1/3 ]]

The determinant of this matrix can be computed directly as det(C) = 1(1/3) - (1/2)(1/2) = 1/3 - 1/4 = 1/12.

Using the explicit Cauchy determinant formula: det(C) = (\prod_(i < j) (x_j - x_i)(y_j - y_i))/(\prod_(i,j) (x_i + y_j)) For our 2 × 2 matrix, the numerator is (x_2 - x_1)(y_2 - y_1) = (2 - 1)(1 - 0) = 1. The denominator is (x_1+y_1)(x_1+y_2)(x_2+y_1)(x_2+y_2) = 1 · 2 · 2 · 3 = 12. The result is 1/12, matching exactly.

Experiment

In the Cauchy Matrix Determinant interactive module, you can investigate larger 3 × 3 matrices by selecting different sets of nodes and applying an optional X shift.

Try the following:

1. Observe the determinant of the "Hilbert" preset, which forms a specific case of a Cauchy matrix where x_i = i and y_j = j-1.
2. Apply an X shift to the "Base" nodes and see how the determinant changes but remains calculable using the same formula.
3. Select the "Repeated row" case to see the determinant evaluate to zero, producing a rank 2 matrix due to linearly dependent rows.

Limitations

The visualization focuses on specific presets of nodes up to dimension 3 × 3 to clearly demonstrate the relationships. It requires all node pairs to satisfy x_i + y_j != 0 so the denominators do not evaluate to zero. This module illustrates structured reciprocal matrices, which should not be confused with the Cauchy distribution or the Cauchy-Binet formula for the determinant of matrix products.

References

Further reading on the properties and formulas of Cauchy matrices:

• "A note on the Cauchy determinant" (arXiv:math/9902004)