Matrix Determinants and Inverses

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A Useful Shortcut for Computing (Some) Matrix Determinants and Inverses Aaron M Montgomery Department of Mathematics and Computer Science Baldwin Wallace University Berea, Ohio 44017 [email protected] http://homepages.bw.edu/∼amontgom/ August 4, 2016 Determinants are hard Shortcuts for Determinants / Inverses Example A. Montgomery 2 2 1 1 1 1 1 1 1 1 1 3 Determinants 6 1 2 1 1 1 1 1 1 1 1 7 Inverses 6 7 6 1 1 2 1 1 1 1 1 1 1 7 Generalizations 6 7 6 1 1 1 2 1 1 1 1 1 1 7 6 7 6 1 1 1 1 2 1 1 1 1 1 7 det 6 7 = ? 6 1 1 1 1 1 2 1 1 1 1 7 6 7 6 1 1 1 1 1 1 2 1 1 1 7 6 7 6 1 1 1 1 1 1 1 2 1 1 7 6 7 4 1 1 1 1 1 1 1 1 2 1 5 1 1 1 1 1 1 1 1 1 2 Example What is the determinant of the matrix obtained by adding 1 to every entry of In? Determinants are hard Shortcuts for Determinants / Inverses A. Montgomery Determinants Inverses Example Generalizations What is the determinant of 2 2 3 0 −1 3 2 1 3 0 −1 3 2 1 0 0 0 3 6 2 7 0 −2 7 6 2 6 0 −2 7 6 0 1 0 0 7 6 7 = 6 7 + 6 7 ? 4 3 9 1 −3 5 4 3 9 0 −3 5 4 0 0 1 0 5 4 12 0 −3 4 12 0 −4 0 0 0 1 Is it even useful to decompose this matrix into a sum in this way? Lemma / First example Shortcuts for Determinants / Inverses Theorem A. Montgomery If A is an invertible n × n matrix, and u; v 2 Rn, then Determinants T T −1 Inverses det(A + uv ) = det(A)(1 + v A u): Generalizations Example What is the determinant of the matrix obtained by adding 1 to every entry of In? Solution: Let A = In, and let u and v both be the vector of all ones in Rn. 2 1 3 2 1 1 1 3 2 1 3 4 1 5 1 1 1 = 4 1 1 1 5 ; 1 1 1 4 1 5 = 3 1 1 1 1 1 T T −1 So, det(In + uv ) = det(In)(1 + v In u) = n + 1: Second example Shortcuts for Determinants / Inverses A. Montgomery Theorem n Determinants If A is an invertible n × n matrix, and u; v 2 R , then Inverses T T −1 Generalizations det(A + uv ) = det(A)(1 + v A u): Example What is the determinant of 2 2 3 0 −1 3 2 1 3 0 −1 3 2 1 0 0 0 3 6 2 7 0 −2 7 6 2 6 0 −2 7 6 0 1 0 0 7 6 7 = 6 7 + 6 7 ? 4 3 9 1 −3 5 4 3 9 0 −3 5 4 0 0 1 0 5 4 12 0 −3 4 12 0 −4 0 0 0 1 T Solution: Let A = In, u = 1 2 3 4 , and T T v = 1 3 0 −1 . Then det(In + uv ) = 1 + 3 = 4: Lemma / Example Shortcuts for Determinants / Inverses A. Montgomery Theorem If A is an invertible n × n matrix, and u; v 2 n, then Determinants R Inverses A−1uvT A−1 Generalizations (A + uvT )−1 = A−1 − : 1 + vT A−1u Example What is the inverse of the matrix obtained by adding 1 to every entry of In? Solution: Let A = In, and let u and v both be the vector of all ones n T −1 uvT 1 in R . Then (In + uv ) = In − 1+vT u = In − n+1 all ones : 2 2 1 1 3 2 3=4 −1=4 −1=4 3 For example, 4 1 2 1 5 4 −1=4 3=4 −1=4 5 = I3: 1 1 2 −1=4 −1=4 3=4 Generalizing the lemmas Shortcuts for Determinants / Inverses A. Montgomery Determinants Theorem (Generalized Matrix Determinant Lemma) Inverses If A is an invertible n × n matrix, U; V are n × m matrices, and W is Generalizations an m × m matrix, then det(A + UWVT ) = det(W) det(A) det(W−1 + VT A−1U): Theorem (Woodbury Matrix Identity) If A is an invertible n × n matrix, U; V are n × m matrices, and W is an invertible m × m matrix, then (A + UWVT )−1 = A−1 − A−1U(W−1 + VT A−1U)−1VT A−1: An exercise Shortcuts for Determinants / What are the determinant and inverse of this matrix? Inverses 2 3 A. Montgomery 2 0 1 0 1 0 1 6 0 2 0 1 0 1 0 7 Determinants 6 7 6 1 0 2 0 1 0 1 7 Inverses 6 7 6 0 1 0 2 0 1 0 7 Generalizations 6 7 6 1 0 1 0 2 0 1 7 6 7 4 0 1 0 1 0 2 0 5 1 0 1 0 1 0 2 http://goo.gl/IYQ6bo.

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Package 'Woodburymatrix'
Package ‘WoodburyMatrix’ August 10, 2020 Title Fast Matrix Operations via the Woodbury Matrix Identity Version 0.0.1 Description A hierarchy of classes and methods for manipulating matrices formed implic- itly from the sums of the inverses of other matrices, a situation commonly encountered in spa- tial statistics and related ﬁelds. Enables easy use of the Woodbury matrix identity and the ma- trix determinant lemma to allow computation (e.g., solving linear systems) without hav- ing to form the actual matrix. More information on the underlying linear alge- bra can be found in Harville, D. A. (1997) <doi:10.1007/b98818>. URL https://github.com/mbertolacci/WoodburyMatrix BugReports https://github.com/mbertolacci/WoodburyMatrix/issues License MIT + ﬁle LICENSE Encoding UTF-8 LazyData true RoxygenNote 7.1.1 Imports Matrix, methods Suggests covr, lintr, testthat, knitr, rmarkdown VignetteBuilder knitr NeedsCompilation no Author Michael Bertolacci [aut, cre, cph] (<https://orcid.org/0000-0003-0317-5941>) Maintainer Michael Bertolacci <[email protected]> Repository CRAN Date/Publication 2020-08-10 12:30:02 UTC R topics documented: determinant,WoodburyMatrix,logical-method . .2 instantiate . .2 mahalanobis . .3 normal-distribution-methods . .4 1 2 instantiate solve-methods . .5 WoodburyMatrix . .6 WoodburyMatrix-class . .8 Index 10 determinant,WoodburyMatrix,logical-method Calculate the determinant of a WoodburyMatrix object Description Calculates the (log) determinant of a WoodburyMatrix using the matrix determinant lemma. Usage ## S4 method for signature 'WoodburyMatrix,logical' determinant(x, logarithm) Arguments x A object that is a subclass of WoodburyMatrix logarithm Logical indicating whether to return the logarithm of the matrix. Value Same as base::determinant.
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