• 0 = dI = d (XX^-1) = dX X^-1 + X d (X^-1)
• d(X^-1) = -X^-1dX X^-1

Note that e_ie_j^T is a matrix containing a 1 in position i, j and zeros elsewhere.

This follows straightforwardly from 2.1 and 2.2.

• d/dx_ij (tr(AXB)) = tr(Ae_ie_j^T B) = tr(e_j^T BAe_i) = e_j^T BAe_i = (BA)_ji = (A^TB^T)_ij[2.3]

• d/dx_ij (tr(AX^-1B)) = -tr(AX^-1e_ie_j^T X^-1B) = -tr(e_j^T X^-1BAX^-1e_i) = -e_j^T X^-1BAX^-1e_i = -(X^-1BAX^-1)_ji = -(X^-TA^TB^TX^-T)_ij [2.3]

• d{tr((AXB+C)D(AXB+C)^H)} = tr{(A(dX)B)D(AXB+C)^H} + tr{(AXB+C)D(A(dX)B)^H}
• = tr{A(dX)BD(AXB+C)^H} + tr{((AXB+C)DB^H(dX)^HA^H}
• = tr{BD(AXB+C)^HA(dX)} + tr{A^H(AXB+C)DB^H(dX)^H}   [1.17]
• = (BD(AXB+C)^HA)^H:^HdX: + ((A^H(AXB+C)DB^H):^HdX:)^C   [1.18]
• = ((A^H(AXB+C)DB^H):^HdX:) +  ((A^H(AXB+C)DB^H):^HdX:)^C
• = {(2A^H(AXB+C)DB^H):^H dX:}^R

• d{tr((AXB+C)D(AXB+C)^H)} = 0
• ⇒ {(2A^H(AXB+C)DB^H):^H dX:}^R = 0  [2.6]
• ⇒ (2A^H(AXB+C)DB^H): = 0  since it must be true for any dX
• ⇒ A^H(AXB+C)DB^H = 0  removing the vectorization
• ⇒ A^HAXBDB^H =  -A^HCDB^H
• ⇒  X = -(A^HA)^-1A^HCDB^H(BDB^H)^-1

• ∂{tr((AXB+C)D(AXB+C)^H)+tr(K^H(EXF-G))+tr((EXF-G)^HK)}/∂X = 0
• ⇒ A^H(AXB+C)DB^H) +E^HKF^H = 0    [2.4]
• ⇒ A^HAXBDB^H = -(A^HCDB^H +E^HKF^H )
• ⇒ X = -(A^HA)^-1(A^HCDB^H +E^HKF^H )(BDB^H)^-1
• Substituting this into the constraint, EXF-G=0, gives
• ⇒ -E(A^HA)^-1(A^HCDB^H +E^HKF^H )(BDB^H)^-1F = G
• ⇒  E(A^HA)^-1E^HKF^H (BDB^H)^-1F = -(G + E(A^HA)^-1A^HCDB^H(BDB^H)^-1F)
• ⇒ K = -(E(A^HA)^-1E^H)^-1( E(A^HA)^-1A^HCDB^H(BDB^H)^-1F+G)(F^H (BDB^H)^-1F)^-1
• Finally, substituting this back into the previous expression for X gives
• X = (A^HA)^-1( E^H{E(A^HA)^-1E^H}^-1{ E(A^HA)^-1A^HCDB^H(BDB^H)^-1F+G}{F^H (BDB^H)^-1F}^-1F^H - A^HCDB^H)(BDB^H)^-1

• d/dx_ij (a^TX^-1b) = -a^TX^-1e_ie_j^TX^-1b = -a^TX^-1e_i * e_j^TX^-1b = -e_i^TX^-Ta * b^TX^-Te_j =  -e_i^TX^-Tab^TX^-Te_j
  Hence d/dX (a^TX^-1b) = -X^-Tab^TX^-T [2.3]

• det(X) I = X ADJ(X) so it follows that det(X) =  sum_j(x_ij ADJ(X)_ji) for each i
• d/dx_ij det(X) = ADJ(X)_ji = (ADJ(X)^T)_ij since ADJ(X)_ji does not depend on x_ij
• d(det(X)) = ADJ(X)^T:^T dX:

• d(det(A^TXB)) = ADJ(A^TXB)^T:^T d(A^TXB): [2.10] = ADJ(A^TXB)^T:^T (B ⊗ A)^T d(X): = ((B ⊗ A) ADJ(A^TXB)^T:)^T d(X): = (A ADJ(A^TXB)^TB^T):^T  dX:
• (A ADJ(A^TXB)^TB^T):^T  dX: = [A,B,X: nonsingular] det(A^TXB) × (A (A^TXB)^-TB^T):^T  dX: =  det(A^TXB) × (A A^-1X^-TB^-TB^T):^T  dX: =  det(A^TXB) × (X^-T):^T  dX:

• d(ln(det(A^TXB))) = det(A^TXB)^-1 × d(det(A^TXB)) = [A,B,X: nonsingular] det(A^TXB)^-1 × det(A^TXB) × (X^-T):^T  dX: [2.11] = (X^-T):^T dX:

• d(det(X)^k) =  k × det(X)^k-1 × d(det(X)) =  k × det(X)^k-1 × det(X) × (X^-T):^T dX: [2.10]

• d(det(X^TCX)) = det(X^TCX)×(X^TCX)^-T:^T (d(X^TCX)): [2.10] = det(X^TCX)×(X^TCX)^-T:^T ( (I ⊗ X^TC) dX: + (X^TC^T ⊗ I) dX^T: ) =  det(X^TCX)×( (C^TX (X^TCX)^-T):^T  dX: + ((X^TCX)^-T X^TC^T):^T dX^T: )  =  det(X^TCX)×( (C^TX (X^TC^TX)^-1)   + (CX(X^TCX)^-1) ):^T  dX:  =  [C=C^T]  2det(X^TCX) × (CX(X^TCX)^-1):^T dX:

• d(det(X^HCX)) = det(X^HCX)×(X^HCX)^-T:^T (d(X^HCX)): [2.10] = det(X^HCX)×(X^HCX)^-T:^T ( (I ⊗ X^HC) dX: + (X^TC^T ⊗ I) dX^H: ) =  det(X^HCX)×( (C^TX^C (X^HCX)^-T):^T  dX: + ((X^HCX)^-T X^TC^T):^T dX^H: )  =  det(X^HCX)× (C^TX^C (X^TC^TX^C)^-1):^TdX:   + (CX(X^HCX)^-1):^T dX^C:)

• d(ln(det(X^HCX))) = (det(X^HCX))^-1× d(det(X^HCX)) = (C^TX^C (X^TC^TX^C)^-1):^TdX:   + (CX(X^HCX)^-1):^T dX^C: [2.15]

• (d/dx (d/dx (Ax+b)^HC(Ax+b))^H)^T = (d/dx ((Ax+b)^HCA)^H)^T = (d/dx (A^HC(Ax+b)))^T = (A^HCA)^T