Modification in Gram-Schmidt process

ISTER_REG

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Oct 28, 2020
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Hi,

lets say that the [imath]b_1^*,...,b_n^*[/imath] are Gram Schmidt vectors and let [imath]v[/imath] be a vector in [imath]\mathbb{R}^n[/imath]. What is [imath]\frac{\langle v , b_i^* \rangle}{||b_i^*||^2}[/imath]? How can I interpret this? Is this for example the length of the component of [imath]v[/imath] projected on [imath]b_i^*[/imath]? I have not a very clear idea, but I would be interested to know how to understand the [imath]\frac{\langle v , b_i^* \rangle}{||b_i^*||^2}[/imath] (graphically/analytically).

Thanks for your help!
 
Hi,

lets say that the [imath]b_1^*,...,b_n^*[/imath] are Gram Schmidt vectors and let [imath]v[/imath] be a vector in [imath]\mathbb{R}^n[/imath]. What is [imath]\frac{\langle v , b_i^* \rangle}{||b_i^*||^2}[/imath]? How can I interpret this? Is this for example the length of the component of [imath]v[/imath] projected on [imath]b_i^*[/imath]? I have not a very clear idea, but I would be interested to know how to understand the [imath]\frac{\langle v , b_i^* \rangle}{||b_i^*||^2}[/imath] (graphically/analytically).

Thanks for your help!
No, it is not length of the projection (which is equal to [imath]\frac{<v,b>}{||b||}[/imath]. The only interpretation I can think of is the ratio of the lengths of the projection of [imath]v[/imath] and that of [imath]b[/imath].
 
Ok so the proportion between the side length of the projection of [imath]v[/imath] onto [imath]b[/imath] and [imath]b[/imath] itself?

Why do you use [imath]b[/imath] instead of [imath]b_i^*[/imath] ?
 
I understand what you mean, but [imath]b_i^*[/imath] underlines that we have a Gram-Schmid vector, whereas [imath]b[/imath] is just "a" vector
 
I understand what you mean, but [imath]b_i^*[/imath] underlines that we have a Gram-Schmid vector, whereas [imath]b[/imath] is just "a" vector
What is good for "a" vector should be good for a Gram-Schmid vector. I.e., replacing [imath]b[/imath] with [imath]b^\star_i[/imath] is left as an exercise for the reader -- please let me know if you have difficulty with it :)
 
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