Let $x_1,\cdots,x_n$ be positive real numbers. Let $A$ be the $n\times n$ matrix whose $i,j^\text{th}$ entry is $$a_{ij}=\frac{1}{x_i+x_j}.$$ This is a Cauchy matrix. I am trying to show that this matrix is positive semi-definite. I have been given the following hint: Consider the matrix $...

Cauchy matrix $C$ is defined by $$C_{i,j}=\frac{1}{a_i + b_j}$$ where $a_i$ and $b_j$ are any numbers so long $a_i + b_j \neq 0$. Why did Cauchy introduce this matrix? Did he use it in the context of another problem or application?

Given 2n distinct real numers $s_1,s_2, \dots, s_n$ and $t_1, t_2, \dots,t_n$ define the $n \times n$ Cauchy matrix $C = C(t,s)$ by $C_{ij} = \frac{1}{t_i - s_j}$. Express the Lagrange interpolation formula: $$p(t_i) = \sum_{j=1}^{n}L_j(t_i)f_j$$ where $$L_j(t) = \prod_{k \neq j}\frac{t-s_k}{t_j...

Find the locus of the feet of the perpendicular drawn upon any tangent to the ellipse $\dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} = 1$ from either focus. My Attempt: Let the tangent to the ellipse be $y=mx\pm \sqrt {a^2m^2+b^2}$ Here the slope of tangent is $m$ so the slope of the perpendicular drawn...