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 1 1 1 1 1 1 1 1 1 Determinants 1 2 1 1 1 1 1 1 1 1 Inverses 1 1 2 1 1 1 1 1 1 1 Generalizations 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 det = ? 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 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 3 0 −1 1 3 0 −1 1 0 0 0 2 7 0 −2 2 6 0 −2 0 1 0 0 = + ? 3 9 1 −3 3 9 0 −3 0 0 1 0 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 ∈ 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. 1 1 1 1 1 1 1 1 1 = 1 1 1 , 1 1 1 1 = 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 ∈ R , then
Inverses T T −1 Generalizations det(A + uv ) = det(A)(1 + v A u).
Example What is the determinant of 2 3 0 −1 1 3 0 −1 1 0 0 0 2 7 0 −2 2 6 0 −2 0 1 0 0 = + ? 3 9 1 −3 3 9 0 −3 0 0 1 0 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 ∈ 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 1 1 3/4 −1/4 −1/4 For example, 1 2 1 −1/4 3/4 −1/4 = 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 A. Montgomery 2 0 1 0 1 0 1 0 2 0 1 0 1 0 Determinants 1 0 2 0 1 0 1 Inverses 0 1 0 2 0 1 0 Generalizations 1 0 1 0 2 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 2
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