Matrix Manual: Matrix Calculus

Matrix Reference Manual
Proofs Section 2: Matrix Calculus

2.1	d(X^-1) = -X^-1dX X^-1 0 = dI = d (XX^-1) = dX X^-1 + X d (X^-1) d(X^-1) = -X^-1dX X^-1
2.2	dX/dx_ij = e_ie_j^T where e_i is the i^th column of I. Note that e_ie_j^T is a matrix containing a 1 in position i, j and zeros elsewhere.
2.3	d/dx_ij (X^-1) = -X^-1e_ie_j^T X^-1 = -X^-1e_i(X^-Te_j)^T This follows straightforwardly from 2.1 and 2.2.
2.4	d/dX (tr(AXB)) = A^TB^T 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]
2.5	d/dX (tr(AX^-1B)) = -X^-TA^TB^TX^-T 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]
2.6	d/dX (a^TX^-1b) = -X^-Tab^TX^-T 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]
2.7	d(det(X)) = ADJ(X)^T:^T dX: = [X: nonsingular] det(X) (X^-T):^T dX: 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:
2.8	d(det(A^TXB)) = d(det(B^TX^TA)) = (A ADJ(A^TXB)^TB^T):^T dX: = [A,B: nonsingular] det(A^TXB) × (X^-T):^T dX: d(det(A^TXB)) = ADJ(A^TXB)^T:^T d(A^TXB): [2.7] = 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:
2.9	d(ln(det(A^TXB))) = [A,B: nonsingular] (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: [2.8] = (X^-T):^T dX:
2.10	d(det(X)^k) = k × det(X^k) × (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: [2.7]
2.11	d(det(X^TCX)) = [C=C^T] 2det(X^TCX)×(CX(X^TCX)^-1):^T dX: d(det(X^TCX)) = det(X^TCX)×(X^TCX)^-T:^T (d(X^TCX)): [2.7] = 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:
2.12	d(det(X^HCX)) = det(X^HCX)× ((C^TX^C (X^TC^TX^C)^-1):^TdX: + (CX(X^HCX)^-1):^T dX^C:) d(det(X^HCX)) = det(X^HCX)×(X^HCX)^-T:^T (d(X^HCX)): [2.7] = 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:)
2.13	d(ln(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: [2.12]
2.14	If C=C^H, then H_x (Ax+b)^HC(Ax+b) = (A^HCA)^T (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

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