Given a vector v ∈ R n then show that we can express ∑ k ω k v k 2 as a matrix product of the form v T M v. Give an expression for M in terms of ω = [ ω 1 . . . ω n ] T I understand that here, the product of a row vector, matrix, and column vector (in that order) is a scalar. However, how do I write M in terms of ω = [ ω 1 . . . ω n ] T ?

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Given a vector   v  ∈            R        n   then show that we can express       ∑          k            ω    k        v    k    2   as a matrix product of the form       v    T    M  v. Give an expression for M in terms of   ω  =  [      ω    1    .  .  .      ω    n        ]    T  I understand that here, the product of a row vector, matrix, and column vector (in that order) is a scalar. However, how do I write M in terms of   ω  =  [      ω    1    .  .  .      ω    n        ]    T  ?

brodireo1 · Accepted Answer

Hint: Note that in general, we have      v    T    M  v  =      ∑          i      =      1        n        ∑          j      =      1        n        m          i      j            v    i        v    j    .I would recommend that you verify this for the cases n=2,3. Now, notice that the expression       ∑    k        ω    k        v    k    2   contains no terms of the form   α        v    i        v    j   (for a scalar   α).

Given a vector v in RR^n then show that we can express sum_k omega_k v_k^2 as a matrix product of the form v^T Mv. Give an expression for M in terms of omega=[omega_1...omega_n]^T

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