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9), that is, KD−1 = 0. 28 Matrix Algebra Next, we discuss an important result in matrix theory known as the matrix inversion lemma, also known as the Woodbury matrix formula. 4. Let A, C, and M = C−1 + DA−1 B be nonsingular, then (A + BCD)−1 = A−1 − A−1 B C−1 + DA−1 B PROOF. 28) Then, (A + BCD) Q = (AQ) + (BCDQ) AA−1 − AA−1 BM−1 DA−1 = + BCDA−1 − BCDA−1 BM−1 DA−1 = I + BCDA−1 − B I + CDA−1 B M−1 DA−1 = I + BCDA−1 − B CC−1 + CDA−1 B M−1 DA−1 = I + BCDA−1 − BC C−1 + DA−1 B M−1 DA−1 = I + BCDA−1 − BCMM−1 DA−1 = I + BCDA−1 − BCDA−1 = I In a similar fashion, one can also show that Q(A + BCD) = I.
Properties of matrix operations Commutative Operations A◦B αA = = B◦A Aα A+B = B+A −1 = A−1 A AA Associativity of Sums and Products A + (B + C) = (A + B) + C A (BC) = (AB) C A ◦ (B ◦ C) = (A ◦ B) ◦ C A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C Distributivity of Products A (B + C) = AB + AC A ⊗ (B + C) = A⊗B+A⊗C (A + B) C = AC + BC (A + B) ⊗ C = A⊗C+B⊗C A ◦ (B + C) = A◦B+A◦C = = B◦A+C◦A (B + C) ◦ A (AB) ⊗ (CD) = (A ⊗ C)(B ⊗ D) Transpose of Products (AB)T = BT AT (A ⊗ B)T = AT ⊗ BT (A ◦ B)T = = BT ◦ AT AT ◦ BT Inverse of Matrix Products and Kronecker Products (AB)−1 = B−1 A−1 (A ⊗ B)−1 = (A)−1 ⊗ (B)−1 Reversible Operations AT T ∗ ∗ (A ) = A = A A−1 −1 =A Vectorization of Sums and Products vec (A + B) = vec (A) + vec (B) vec (BAC) = vec (A ◦ B) = CT ⊗ B vec (A) vec(A) ◦ vec(B) inverses.
By setting one of the nodes as having zero potential (the ground node), we want to determine the potentials of the remaining n nodes as well as the current flowing through each link and the voltages across each of the resistors. To obtain the required equations, we need to first propose the directions of each link, select the ground node (node 0), and label the remaining nodes (nodes 1 to n). Based on the choices of current flow and node labels, we can form the node-link incidence matrix [=]n × m, which is a matrix composed of only 0, 1, and −1.