Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If $A$ is an $n\times n$ nonsingular matrix, and $b$ , $c$ are column vectors of length $n$. Is there an algorithm I can use to compute the matrix $W = bc^\top A^{-1}$ in $\frac{2}{3}n^3 + O(n^2)$ flops?

share|cite|improve this question
up vote 3 down vote accepted


  1. Decompose $\mathbf A$ with your favorite decomposition. (e.g. the LU decomposition, $\mathbf P\mathbf A=\mathbf L\mathbf U$).

  2. Solve the system $\mathbf A^\top\mathbf y=\mathbf c$ with your decomposition.

  3. Form the outer product $\mathbf b\mathbf y^\top$.

I leave the details, including the flop counting, to you.

It seems the hints I gave earlier wasn't sufficiently clear, so here's another:

$$\mathbf A^{-\top}=\mathbf P\mathbf L^{-\top}\mathbf U^{-\top}$$

where $\mathbf A^{-\top}$ denotes the inverse of $\mathbf A^\top$.

You should now be able to figure how to use the LU decomposition of $\mathbf A$ to generate the column vector $\mathbf y$.

share|cite|improve this answer
Sorry I was actually looking for an algorithm that is $\frac{2}{3}n^3$+$O(n^2)$ not $\frac{8}{3}n^3$+$O(n^2)$ – Mark Jul 24 '12 at 6:46
As I said, do the flop counting yourself. How much flops does an LU decomposition take? A backsubstitution? Forming an outer product? – J. M. Jul 24 '12 at 6:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.