Hi Prof Caramanis, in this video you defined the dot product as v1^T * M * v1. Is this based on a general formula of inner product of two matrices? thanks!
Yes, exactly. I am using the fact that the inner product of two matrices A and B of the same size is = \sum_{ij} A_ij B_ij, which also is equivalent to the trace formula you gave.
For copositive matrices exercise. Q1: intuitively, copositive matrices (M) are smaller (subspace) than PSD matrices (S) because x^TSx >= 0 for any x in R^n. But x^TMx >= 0 only for x in R^n_+. Q2: It is similar to the proof of PSD matrices are convex set. Hi Prof Caramanis, I am not familiar with the definition of "given support", and thus, I didn't finish the exercise for "the set of moments of distribution with given support". Could you explain the jargon "support on C" meaning in 6:46? Thanks!
By "moments with a given support" i mean that these are moments of a distribution, where the distribution only puts weight on the set C. So if C is the discrete points {-1,1}, then we are talking about the set of moments of distributions that put weight only on the two points -1 and 1.
Was there a previous course? He began talking about probability distributions and I don't think I have heard that previously
Hi Prof Caramanis, do you plan to release any exercise sets to supplement this lecture series? Thanks!
I'd like to at some point, but no plans to do that in the very immediate future, unfortunately...
Hi Prof Caramanis, in this video you defined the dot product as v1^T * M * v1. Is this based on a general formula of inner product of two matrices? thanks!
Are you using, for two matrices A, B, = Trace(A^T * B) as the inner product formula? thanks
Yes, exactly. I am using the fact that the inner product of two matrices A and B of the same size is = \sum_{ij} A_ij B_ij, which also is equivalent to the trace formula you gave.
@@constantine.caramanis thanks for clarifying! it makes sense.
For copositive matrices exercise. Q1: intuitively, copositive matrices (M) are smaller (subspace) than PSD matrices (S) because x^TSx >= 0 for any x in R^n. But x^TMx >= 0 only for x in R^n_+. Q2: It is similar to the proof of PSD matrices are convex set. Hi Prof Caramanis, I am not familiar with the definition of "given support", and thus, I didn't finish the exercise for "the set of moments of distribution with given support". Could you explain the jargon "support on C" meaning in 6:46? Thanks!
By "moments with a given support" i mean that these are moments of a distribution, where the distribution only puts weight on the set C. So if C is the discrete points {-1,1}, then we are talking about the set of moments of distributions that put weight only on the two points -1 and 1.
@@constantine.caramanis Thank you for your explanation!