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

Regards,
Olof

Kerwin

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.

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

Kerwin

Cheers guys.

Olof