Determinant of Cauchy

In Linear algebra, the determinant of Cauchy is a traditional calculation of determinant, which can be connected to problems of rational fractions. Its name is a homage to the mathematician Augustin Louis Cauchy.

The determinant of Cauchy is determining of size N and of general term $\ frac1 {a_i+b_j}$ , where the complexes a₁,…, a_n and b₁,…, b_n are such as for all I and J , a_i+b_j is nonnull

D_n = \ begin {vmatrix} \ frac1 {a_1+b_1} & \ frac1 {a_1+b_2} & \ dowries & \ frac1 {a_1+b_n} \ \

\ frac1 {a_2+b_1} & \ frac1 {a_2+b_2} & \ dowries & \ frac1 {a_2+b_n} \ \ \ vdots& \ vdots& & \ vdots \ \ \ frac1 {a_n+b_1} & \ frac1 {a_n+b_2} & \ dowries & \ frac1 {a_n+b_n} \ end {vmatrix}

Bond with a problem of interpolation

One seeks a rational fraction having exactly simple N poles, which are the a_i , and taking values fixed in N points distinct from the a_i (they are the opposites of the b_j ).

If one seeks the rational fraction in the form

F (X) = \ sum_ {i=1} ^n \ frac {r_i} {X-a_i}

then the unknown coefficients r_i are solutions of a system of size N , of determinant a determinant of Cauchy.

Calculation of the determinant of Cauchy

$D_n= \ frac {\ prod \ limits_ {i$

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