(a) Prove that
∇x(xTa) = ∇x(aTx) = a
for any two n-dimensional column vectors x, a. Hint: differentiate w.r.t. each element of x, and then gather the partial derivatives into a column vector.
(1)
(2)
(3)
(4)
(b) Prove that
∇x(xTAx) = (A + AT)x
for any column vector x, any n × n matrix A, and any constant ndimensional vector b.
1
n
= x1Ak,1 + ... + (xk XAi,kxi + Ak,kxk) + ... + xnAk,n
i=1
(15)
n
= (x1Ak,1 + ... + xnAk,n) + xk XAi,kxi
i=1
(16)
n n
= XAk,ixi + XAi,kxi
i=1 i=1
(17)
= (Ak,. + AT.,k)x 2
(A1,. + AT.,1)x
∇x(xTAx) = ...
(An,. + AT.,n)x
(A1,. (AT.,1
.
= .. + ... x
An,. AT.,n
= (A + AT)x
(18)
(19)
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(21)
(c) Based on the theorem above, prove that
∇x(xTAx) = 2Ax
for any n-dimensional column vector x and any symmetric n × n matrix A.
∇x(xTAx) = (A + AT)x (22)
= (A + A)x (23)
= 2Ax (24)
(d) Based on the theorems above, prove that
for any n-dimensional column vector x, any symmetric n × n matrix A, and any constant n-dimensional column vector b.
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