Contents - Student subdomain for University of Bath

230 APPENDIX B. EXCURSUS
many columns), this would show that the complexity of matrix inverse is also
O(n ω ) if ω > 2 (if ω = 2 we get an extra factor of log n).
Lemma 16 If B is nonsingular then A := B T B is symmetric and positive
definite (for any non-zero x, xAx T > 0).
Lemma 17 If A is symmetric and positive definite then, in the notation of
Lemma 15, so are P and U.
Theorem 40 If B is an invertible square matrix, B −1 can be computed in time
O(n ω ) if ω > 2 (if ω = 2 we get an extra factor of log n).
Proof. We compute A := B T B in time O(n ω ), its inverse by the process after
Lemma 15, and then B −1 := A −1 A T (another O(n ω )).