Woodbury matrix identity

Woodbury matrix identity   (Redirected from Binomial inverse theorem) Jump to navigation Jump to search In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury,[1][2] says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.[3][4] The Woodbury matrix identity is[5] {displaystyle left(A+UCVright)^{-1}=A^{-1}-A^{-1}Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1},} where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion.

While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category.

Contents 1 Discussion 1.1 Special cases 1.1.1 Inverse of a sum 1.2 Variations 1.2.1 Binomial inverse theorem 2 Derivations 2.1 Direct proof 2.2 Alternative proofs 3 Applications 4 See also 5 Notes 6 External links Discussion To prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrix I, we obtain another identity which is a bit simpler: {displaystyle left(I+UVright)^{-1}=I-Uleft(I+VUright)^{-1}V.} To recover the original equation from this reduced identity, set {displaystyle U=A^{-1}X} and {displaystyle V=CY} .

This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from {displaystyle I=(I+P)^{-1}(I+P)=(I+P)^{-1}+(I+P)^{-1}P} , thus, {displaystyle (I+P)^{-1}=I-(I+P)^{-1}P} , and similarly {displaystyle (I+P)^{-1}=I-P(I+P)^{-1}.} The second identity is the so-called push-through identity[6] {displaystyle (I+UV)^{-1}U=U(I+VU)^{-1}} that we obtain from {displaystyle U(I+VU)=(I+UV)U} after multiplying by {displaystyle (I+VU)^{-1}} on the right and by {displaystyle (I+UV)^{-1}} on the left.

Special cases When {displaystyle V,U} are vectors, the identity reduces to the Sherman–Morrison formula.

In the scalar case it (the reduced version) is simply {displaystyle {frac {1}{1+uv}}=1-{frac {uv}{1+uv}}.} Inverse of a sum If n = k and U = V = In is the identity matrix, then {displaystyle {begin{aligned}left({A}+{B}right)^{-1}&=A^{-1}-A^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}\&={A}^{-1}-{A}^{-1}left({A}{B}^{-1}+{I}right)^{-1}.end{aligned}}} Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity {displaystyle left({A}+{B}right)^{-1}={A}^{-1}-left({A}+{A}{B}^{-1}{A}right)^{-1}.} Another useful form of the same identity is {displaystyle left({A}-{B}right)^{-1}={A}^{-1}+{A}^{-1}{B}left({A}-{B}right)^{-1},} which has a recursive structure that yields {displaystyle left({A}-{B}right)^{-1}=sum _{k=0}^{infty }left({A}^{-1}{B}right)^{k}{A}^{-1}.} This form can be used in perturbative expansions where B is a perturbation of A.

Variations Binomial inverse theorem If A, B, U, V are matrices of sizes n×n, k×k, n×k, k×n, respectively, then {displaystyle left(A+UBVright)^{-1}=A^{-1}-A^{-1}UBleft(B+BVA^{-1}UBright)^{-1}BVA^{-1}} provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals B(I + VA−1UB) and the rank of the latter cannot exceed the rank of B.[6] Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in the original Woodbury identity.

A variation for when B is singular and possibly even non-square:[6] {displaystyle (A+UBV)^{-1}=A^{-1}-A^{-1}U(I+BVA^{-1}U)^{-1}BVA^{-1}.} Formulas also exist for certain cases in which A is singular.[7] Derivations Direct proof The formula can be proven by checking that {displaystyle (A+UCV)} times its alleged inverse on the right side of the Woodbury identity gives the identity matrix: {displaystyle {begin{aligned}&left(A+UCVright)left[A^{-1}-A^{-1}Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}right]\={}&left{I-Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}right}+left{UCVA^{-1}-UCVA^{-1}Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}right}\={}&left{I+UCVA^{-1}right}-left{Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}+UCVA^{-1}Uleft(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}right}\={}&I+UCVA^{-1}-left(U+UCVA^{-1}Uright)left(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}\={}&I+UCVA^{-1}-UCleft(C^{-1}+VA^{-1}Uright)left(C^{-1}+VA^{-1}Uright)^{-1}VA^{-1}\={}&I+UCVA^{-1}-UCVA^{-1}\={}&I.end{aligned}}} Alternative proofs show Algebraic proof show Derivation via blockwise elimination show Derivation from LDU decomposition Applications This identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute (A + UCV)−1. With the inverse of A available, it is only necessary to find the inverse of C−1 + VA−1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A + UCV directly. A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.

This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter.

In the case when C is the identity matrix I, the matrix {displaystyle I+VA^{-1}U} is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.[4] See also Sherman–Morrison formula Schur complement Matrix determinant lemma, formula for a rank-k update to a determinant Invertible matrix Moore–Penrose pseudoinverse#Updating the pseudoinverse Notes ^ Max A. Woodbury, Inverting modified matrices, Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950, 4pp MR38136 ^ Max A. Woodbury, The Stability of Out-Input Matrices. Chicago, Ill., 1949. 5 pp. MR32564 ^ Guttmann, Louis (1946). "Enlargement methods for computing the inverse matrix". Ann. Math. Statist. 17 (3): 336–343. doi:10.1214/aoms/1177730946. ^ Jump up to: a b Hager, William W. (1989). "Updating the inverse of a matrix". SIAM Review. 31 (2): 221–239. doi:10.1137/1031049. JSTOR 2030425. MR 0997457. ^ Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. p. 258. ISBN 978-0-89871-521-7. MR 1927606. ^ Jump up to: a b c Henderson, H. V.; Searle, S. R. (1981). "On deriving the inverse of a sum of matrices" (PDF). SIAM Review. 23 (1): 53–60. doi:10.1137/1023004. hdl:1813/32749. JSTOR 2029838. ^ Kurt S. Riedel, "A Sherman–Morrison–Woodbury Identity for Rank Augmenting Matrices with Application to Centering", SIAM Journal on Matrix Analysis and Applications, 13 (1992)659-662, doi:10.1137/0613040 preprint MR1152773 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.7.3. Woodbury Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 External links Some matrix identities Weisstein, Eric W. "Woodbury formula". MathWorld. Categories: Lemmas in linear algebraMatricesMatrix theory