- #1

Peter_Newman

- 155

- 11

Hi,

The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...]

So the cosine is the ratio of the adjacent to the hypotenuse. That means between ##b_i^*## and ##b_i## there is always this ratio, I would accept that. But what irritates me a bit is the statement about the angle.

I have drawn this now for the case ##i=2## and I'm the opinion that what stands above is not completely correct, correctly would be, if it would be called ##\theta_i## is the angle between ##b_i## and ##span(b_1,...,b_{i-1})^{\perp}##.

For notation: ##b_i^*## are Gram Schmidt vectors.

What do you think?

The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...]

So the cosine is the ratio of the adjacent to the hypotenuse. That means between ##b_i^*## and ##b_i## there is always this ratio, I would accept that. But what irritates me a bit is the statement about the angle.

I have drawn this now for the case ##i=2## and I'm the opinion that what stands above is not completely correct, correctly would be, if it would be called ##\theta_i## is the angle between ##b_i## and ##span(b_1,...,b_{i-1})^{\perp}##.

For notation: ##b_i^*## are Gram Schmidt vectors.

What do you think?

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