Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula

Leo Giugiuc has kindly communicated to me a problem from an artofproblemsolving forum. The problem is by the forum member kilua; the solution by Claudia Nanuti, Diana Trailescu, Dan Sitaru and Leo Giugiuc.

Problem

Cauchy-Binet formula for prving inequalities

Cauchy-Binet Formula

Given a $k\times n\;$ matrix $A\;$, with $k\le n\;$ and an $n\times k\;$ matrix B. Let $J=(j_1,j_2,\ldots,j_k),\;$ $j_1\le j_2\lt \ldots\lt j_k\le n,\;$ be an ordered subset of $k\;$ out of $n\;$ indices. Define $A_J\;$ as the matrix obtained from $A\;$ by removing all columns that are not in $J\;$ and $B^J\;$ as the matrix obtained from $B\;$ by removing all rows that are not in $J.\;$ Then

Cauchy-Binet formula

$\displaystyle \det(AB)=\sum_{J}\det(A_J)\det(B^J),$

where the sum is taken over all subsets $J\subset\{1,2,\ldots,n\},\;$ $|J|=k.$

Corollary

For a $n\times k\;$ matrix $B,\;$ with $k\le n,$

$\displaystyle \det(B^tB)=\sum_J\left(\det(B^J)\right)^2.$

Solution of the problem

We shall prove the following equivalent inequality:

$\displaystyle\left(\sum_{i=1}^{n}\frac{x_i^2}{a_i}\right)\left(\sum_{i=1}^{n}a_ix_i^2\right)\left(\sum_{i=1}^{n}a_iy_i^2\right)\ge \left(\sum_{i=1}^{n}\frac{x_i^2}{a_i}\right)\left(\sum_{i=1}^{n}a_ix_iy_i\right)^2+\sum_{i=1}^na_iy_i^2.$

Consider matrix $B=(b_{ij})_{1\le i\le n}^{1\le j\le 3},$ where $b_{i1}=\frac{x_i}{\sqrt{a_i}},\;$ $b_{i2}=x_i\sqrt{a_i},\;$ $b_{i3}=y_i\sqrt{a_i},\;$ $i=1,2,\ldots,n.\;$ Then

$\displaystyle B^tB=\left(\begin{array}{ccc} \sum_{i=1}^n\frac{x_i^2}{a_i} & 1 & 0\\ 1 & \sum_{i=1}^na_ix_i^2 & \sum_{i=1}^na_ix_iy_i\\ 0 & \sum_{i=1}^na_ix_iy_i & \sum_{i=1}^na_iy_i^2 \end{array}\right).$

The result follows immediately by the Cauchy-Binet formula (and a verification by the Cauchy-Schwarz inequality).

Thus

$\displaystyle\left(\sum_{i=1}^{n}\frac{x_i^2}{a_i}\right)\left(\sum_{i=1}^{n}a_ix_i^2\right)\left(\sum_{i=1}^{n}a_iy_i^2\right)\ge \left(\sum_{i=1}^{n}\frac{x_i^2}{a_i}\right)\left(\sum_{i=1}^{n}a_ix_iy_i\right)^2+\sum_{i=1}^na_iy_i^2.$

The equality holds for $n=2.\;$ Indeed, $rank(B)=rank(B^t)\le 2,\;$ such that $rank(B^tB)\le 2,\;$ implying $\det(B^tB)=0.$

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Linear Algebra Tools for Proving Inequalities

$\;\left(\displaystyle\left(\frac{a}{b-c}\right)^2+\left(\frac{b}{c-a}\right)^2+\left(\frac{c}{a-b}\right)^2\ge 2\right)$
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