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.