Non-negative: Yes, square of a number is non-negative, therefore a sum of non-negative numbers is still non-negative, taking a square root also does not change that
(this one might be a bit hand wavey..)
but intuitively, the only way for a sum of non-negative numbers to be zero is if they are all zero (there are no negatives to cancel out the positives)
For each component v_i, the constant factor is squared then square-rooted, so it is the same as multiplying the norm by the constant
Essentially we're comparing:
sqrt((u1 + v1)^2 + (u2 + v2)^2 + ...)
and
sqrt(u1^2+u2^2+...) + sqrt(v1^2+v2^2+...)
we can expand by squaring both sides, and we see that the result on the top is larger, you will have a series of u1^2+u2^2+...+v1^2+v2^2... terms that cancel out
keenan
@genericname Terrific. I don't think 2 is too hand-wavey; perhaps what you are looking for is something like, "if even a single coordinate is nonzero, then the sum is already positive."
Min
What does the ":=" on the right hand side of the top equation mean?
Non-negative: Yes, square of a number is non-negative, therefore a sum of non-negative numbers is still non-negative, taking a square root also does not change that
(this one might be a bit hand wavey..) but intuitively, the only way for a sum of non-negative numbers to be zero is if they are all zero (there are no negatives to cancel out the positives)
For each component v_i, the constant factor is squared then square-rooted, so it is the same as multiplying the norm by the constant
Essentially we're comparing: sqrt((u1 + v1)^2 + (u2 + v2)^2 + ...) and sqrt(u1^2+u2^2+...) + sqrt(v1^2+v2^2+...) we can expand by squaring both sides, and we see that the result on the top is larger, you will have a series of u1^2+u2^2+...+v1^2+v2^2... terms that cancel out
@genericname Terrific. I don't think 2 is too hand-wavey; perhaps what you are looking for is something like, "if even a single coordinate is nonzero, then the sum is already positive."
What does the ":=" on the right hand side of the top equation mean?
@Min it means 'is defined as'