16
$\begingroup$

Suppose $x=(x_1,x_2),y = (y_1,y_2) \in \mathbb{R}^2$. I noticed that \begin{align*} \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= x_1^2y_1^2 + x_1^2 y_2^2 + x_2^2 y_1^2 + x_2^2 y_2 ^2 - (x_1^2 y_1^2 + 2 x_1 y_1 x_2 y_2 + x_2^2 y_2^2) \\ &=(x_1 y_2)^2 - 2x_1 y_2 x_2 y_1 + (x_2 y_2)^2 \\ &=(x_1 y_2 - x_2 y_1)^2 \end{align*} which proves the CSB inequality in dimension two. This begs the question:

If $x = (x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n$, is there a polynomial $p \in \mathbb{R}[x_1,\ldots,x_n;y_1,\ldots,y_n]$ such that $ \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 = p^2$?

$\endgroup$
4
  • 1
    $\begingroup$ I don't know the answer, but I strongly recommend having a look at Steele's book The Cauchy-Schwarz Master Class. link $\endgroup$ Mar 25, 2013 at 18:12
  • $\begingroup$ Apparently CSB Inequality = Cauchy-Schwarz-Buniakovski Ineq. ? $\endgroup$
    – DonAntonio
    Mar 25, 2013 at 18:23
  • $\begingroup$ @DonAntonio Yeah. $\endgroup$
    – Pedro
    Mar 25, 2013 at 18:42
  • 3
    $\begingroup$ This is related to Hilbert's 17th problem. Artin showed that every nonnegative-valued polynomial is a finite sum of rational squares, hence "obviously" nonnegative. $\endgroup$
    – Erick Wong
    Mar 25, 2013 at 18:46

4 Answers 4

15
$\begingroup$

No (it is not a square of a polynomial for $n \ge 3$), but the right generalization, proving that it is nonnegative, is that it is a sum of squares.

For instance, for $n = 3$, $$ \begin{align*} \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= (x_1y_2 - x_2y_1)^2 + (x_2y_3-x_3y_2)^2 + (x_3y_1 - x_1y_3)^2, \end{align*} $$

and in general $$ \begin{align*} \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= \sum_{i < j}(x_iy_{j} - x_{j}y_{i})^2 \end{align*} $$


This is easy to prove algebraically: the left hand side is

$$ \begin{align*} \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= (\sum{x_i^2}\sum{y_j^2}) - (\sum{x_iy_i})^2 \\ &= \sum_{i=j}{x_i^2 y_j^2} + \sum_{i\neq j}{x_i^2y_j^2} - \sum_{i=j}{x_iy_ix_iy_i} - \sum_{i\neq j}{x_iy_ix_jy_j} \\ &= \sum_{i<j}{(x_i^2 y_j^2 + x_j^2 y_i^2)} - \sum_{i<j}{2x_iy_ix_jy_j} \\ &= \sum_{i<j}{(x_i^2 y_j^2 - 2x_iy_jx_jy_i + x_j^2y_i^2)} \\ &= \sum_{i < j}(x_iy_{j} - x_{j}y_{i})^2 \end{align*} $$

This identity is known as Lagrange's identity.


This also shows that the left hand size is zero when for all pairs $(i,j)$, we have $x_iy_j - x_jy_i = 0$, i.e., $$\frac{y_i}{x_i} = \frac{y_j}{x_j}$$ (let's assume the $x_j$s are nonzero, for now), which is another way of saying that one vector is a multiple of the other, i.e., equality holds in the inequality when the two vectors are parallel.


For the former (showing that it is not the square of a polynomial), consider for instance $n=3$. If $\|x\|^2 \|y\|^2 - \langle x,y \rangle^2$ is the square of a polynomial $p(x_1, x_2, x_3, y_1, y_2, y_3)$, then we can write the polynomial as $qx_1 + r$, where $q = q(x_2, x_3, y_1, y_2, y_3)$ and $r = r(x_2, x_3, y_1, y_2, y_3)$ are polynomials that don't depend on $x_1$. As $(qx_1+r)^2 = q^2x_1^2 + 2qrx_1 + r^2$, the coefficient of $x_1^2$ should be a square, but the coefficient is $y_1^2 + y_2^2 + y_3^2 - y_1^2 = y_2^2 + y_3^2$ (or in general, $\sum_{i=2}^{n}y_i^2$), which is not the square of a polynomial. (Proved similarly: if it is the square of $qy_2 + r$, then comparing coefficients of $y_2^2$ gives $q \equiv 1$, and comparing coefficents of $y_2$ gives $r \equiv 0$, which is not consistent with the rest.)


As Erick Wong points out in the comments, this is related to (the solution of) Hilbert's seventeenth problem, which says that any polynomial that takes only nonnegative values can be written as a sum of squares of rational functions. If we only care about representations as sum of squares of polynomials, any nonnegative polynomial can be approximated as closely as desired with a sum of squares of polynomials. See e.g. the book Positive Polynomials and Sums of Squares (preview).

$\endgroup$
4
  • 1
    $\begingroup$ Are you sure? I don't understand why terms like $x_i y_j$ for $\vert j - i \vert > 1$ wouldn't appear in the general form for $n > 3$. $\endgroup$
    – roger
    Mar 25, 2013 at 18:34
  • $\begingroup$ @roger: Sorry, you are right... was fixing it when I saw your comment. $\endgroup$ Mar 25, 2013 at 18:36
  • $\begingroup$ ok now I find it plausible :). $\endgroup$
    – roger
    Mar 25, 2013 at 18:47
  • $\begingroup$ Ah of course! Thanks this is very nice to know. I think when I first saw the CSB inequality it was in the context of Hilbert spaces. I had no idea there was such a nice simple algebraic proof in finite dimension. $\endgroup$
    – Mike F
    Mar 25, 2013 at 22:26
5
$\begingroup$

$\sum x_i^2\sum y_i^2-(\sum x_iy_i)^2$

$=\sum x_i^2y_i^2+\sum_{i\neq j} x_i^2y_j^2-\sum x_i^2y_i^2-\sum_{i\neq j} x_iy_ix_jy_j$

$=\sum_{i<j} (x_iy_j-x_jy_i)^2$

$\endgroup$
3
  • 1
    $\begingroup$ Some words never hurt nobody! $\endgroup$
    – Pedro
    Mar 25, 2013 at 18:42
  • 2
    $\begingroup$ @PeterTamaroff Try that without the double negative. $\endgroup$
    – user14082
    Mar 25, 2013 at 19:07
  • 2
    $\begingroup$ Double, triple and more negative are advanced, nice features of well developed and beautiful languages, like spanish, russian and wooly-wooly (which, if it existed, would be widely known and spoken in some 1,000 tiny islands in Micronesia). In fact, it should have been: "Don't some words never didn't hurt anybody?" $\endgroup$
    – DonAntonio
    Mar 25, 2013 at 23:16
5
$\begingroup$

When $n\ge3$, $\|x\|^2\|y\|^2 - \langle x,y\rangle^2$ is not the square of any polynomial $p(x,y)$. Keep all entries other than $x_1$ fixed and let \begin{align*} q(x_1) &= \|x\|^2\|y\|^2 - \langle x,y\rangle^2,\\ \Rightarrow q\,'(x_1) &= 2(x_1 \|y\|^2 - y_1\langle x,y\rangle) \end{align*} If $q$ is a squared polynomial, some zero of $q\,'$ must be a zero of $q$. However, when $x=(x_1,1,0,0,\ldots,0)$ and $y=(1,0,1,0,0,\ldots,0)$, we have $q\,'(x_1)=2x_1$ and $q(x_1)=2(x_1^2+1)-x_1^2$. So, the only zero of $q\,'$ is $x_1=0$, but $q(0)=2\neq0$. Therefore $q$ is not a squared polynomial.

$\endgroup$
3
$\begingroup$

CSB inquality can be generalized: Gramian's matrix determinant is equal or greater than a zero. And equality means linear dependence of system of vectors. In case of two vectors $u$ and $v$, we have:

$$ 0 \leqslant \begin{vmatrix} \langle u,u \rangle & \langle u,v \rangle \\ \langle v,u \rangle & \langle v,v \rangle \\ \end{vmatrix} = \|u\|^2 \cdot \|v\|^2 - \langle u,v \rangle^2 $$

So, it's another proof of CSB.

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .