Cauchy-Schwarz inequality

By Olof Sisask (P3033) on Sunday, April 22, 2001 - 06:38 pm :

What is Cauchy-Schwarz? I've seen it mentioned a couple of times on here, but never asked about it.

Regards,
Olof

By Kerwin Hui (Kwkh2) on Sunday, April 22, 2001 - 06:44 pm :

C-S states that

{(\sum x_{j} y_{j})}^{2} \leq (\sum x_{j}^{2}) (\sum y_{j}^{2})

with equality if and only if

x

and

y

are parallel, i.e. there exists

λ

and

μ

, not both zero, such that

λ x_{i} = μ y_{i}

, all

i

Kerwin

By Michael Doré (Michael) on Wednesday, April 25, 2001 - 02:56 pm :

This is basically an extension of the fact that the magnitude of a cosine of a real angle is less than or equal to 1.

If:

x = (x₁ ,x₂ )
y = (y₁ ,y₂ )

Then (x₁ y₁ + x₂ y₂ )² = (x .y )² = |x |² |y |² cos² (theta) < = |x |² |y |² = (x₁ ² + x₂ ² )(y₁ ² + y₂ ² )

where theta is the angle between x and y . This demonstrates the inequality for n = 2 and also makes it clear why they need to be parallel for equality. You can justify it in a similar way for N-D though you need to be careful about how you define angle. Alternatively, you can prove the inequality by expanding it out and collecting terms and using the fact that squares are positive.

By Kerwin Hui (Kwkh2) on Wednesday, April 25, 2001 - 10:23 pm :

Here is a previous thread where C-S was discussed about halfway down the page.

Kerwin

By Olof Sisask (P3033) on Wednesday, April 25, 2001 - 10:43 pm :

Cheers guys.

Olof