General inverse for a symmetric matrix of the form $\mathbf A_n = \left(a_{\min(i,j)}b_{\max(i,j)}\right)$ I am looking for a general inverse formula for a symmetric matrix of the form
$$ \mathbf A_n = \left(a_{\min(i,j)}b_{\max(i,j)}\right) = \pmatrix{ a_1b_1 & a_1b_2 & a_1b_3 & \cdots & a_1b_n \\
a_1b_2 & a_2b_2 & a_2b_3 & \cdots & a_2b_n \\
\vdots &    & \ddots  &        & \vdots \\
a_1b_n & a_2b_n &   &        & a_nb_n} $$
where $a_i,b_i $ are real numbers.  Empirically I can see that $ \mathbf A_n^{-1} $ is tridiagonal so I suspect it has previously been studied.  For example we have
$$ \mathbf A_5 =
\left(
\begin{array}{ccccc}
 \frac{a_2}{a_1 a_2 b_1-a_1^2 b_2} & \frac{1}{a_1 b_2-a_2 b_1} & 0 & 0 & 0 \\
 \frac{1}{a_1 b_2-a_2 b_1} & \frac{a_1 b_3-a_3 b_1}{\left(a_2 b_1-a_1 b_2\right) \left(a_2 b_3-a_3 b_2\right)} & \frac{1}{a_2 b_3-a_3 b_2} & 0 & 0 \\
 0 & \frac{1}{a_2 b_3-a_3 b_2} & \frac{a_2 b_4-a_4 b_2}{\left(a_3 b_2-a_2 b_3\right) \left(a_3 b_4-a_4 b_3\right)} & \frac{1}{a_3 b_4-a_4 b_3} & 0 \\
 0 & 0 & \frac{1}{a_3 b_4-a_4 b_3} & \frac{a_3 b_5-a_5 b_3}{\left(a_4 b_3-a_3 b_4\right) \left(a_4 b_5-a_5 b_4\right)} & \frac{1}{a_4 b_5-a_5 b_4} \\
 0 & 0 & 0 & \frac{1}{a_4 b_5-a_5 b_4} & \frac{b_4}{a_5 b_4 b_5-a_4 b_5^2} \\
\end{array}
\right)
$$
which seems to be based on a bunch of 2x2 determinants...
Any help appreciated, thank you!
p.
 A: Ok so after researching inversion of symmetric tridiagonal matrices, it appears that $ \mathbf A$ is known as a "matrice factorisable" with tridiagonal inverse
$$
 \mathbf A^{-1} =
\pmatrix{ \alpha_1 & \beta_1 & & \\
\beta_1 & \alpha_2 & \beta_2 & \\
& \ddots & \ddots & \\
& & \beta_{n-1} & \alpha_n }
$$
where:
$$
\alpha_1 = - \frac{a_2}{a_1 u_{1,2}},
\alpha_i = \frac{u_{i-1,i+1}}{u_{i-1,i}u_{i,i+1}},
\alpha_n = -\frac{b_{n-1}}{b_n u_{n-1,n}},
\beta_i = \frac{1}{u_{i,i+1}},
u_{i,j} = a_i b_j (v_j - v_i),
v_i = \frac{b_i}{a_i}
$$
subject to a's and b's being nonzero and $v_i \neq v_{i+1}$.
There is also an interesting factorization of the form $\mathbf A = \mathbf {DL}^T\boldsymbol\Lambda\mathbf{LD}$ where $\mathbf D$ is diagonal, $\mathbf L$ is lower triangular, and $\boldsymbol\Lambda$ is superdiagonal together with one last diagonal coefficient.  I hope this helps anyone encountering a similar type of matrix!
p.
Reference: J. BARANGER, M. DUC-JACQUET, "Matrices tridiagonales symétriques et matrices factorisables". In: Revue française d’informatique et de recherche opérationnelle, vol. 5, no R3 (1971), pp. 61-66.
