Cauchy Matrix Determinant

A Cauchy matrix is an n × n matrix C constructed from two sequences of values, x_1, ..., x_n and y_1, ..., y_n. The entries of the matrix are defined as C_ij = 1 / (x_i + y_j). As long as x_i + y_j != 0 for all i,j, the matrix is well-defined.

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.