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GAXPY operations

  1. Sep 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Let A be an nxn matrix belonging to R and x be a vector of length k belonging to R. Find the first column of
    M = (A − x1I)(A − x2I)...(A − xrI) using a sequence of GAXPY’s operations.

    2. Relevant equations
    GAXPY: General matrix A multiplied by a vector X plus a vector Y

    3. The attempt at a solution
    I tried to figure it out by writing A and x explicitly and then multiplying (A-x1I) ...(A-xrI) but it was so messy and i did not get any result, i want a hint how to start the solution.
     
  2. jcsd
  3. Sep 28, 2015 #2

    Mark44

    Staff: Mentor

    The matrix and vector don't "belong to" R. I think what you mean is that the entries in A and x are real numbers. The matrix would be an element of ##\mathbb{R}^{n x n}## and the vector would be (I think) an element of ##\mathbb{R}^k##, unless by "length" you mean its magnitude.
    What is r in the equation above? In other words, how many factors are there on the right?

     
  4. Sep 28, 2015 #3
    what you thought is absolutely right, A is a matrix in R^(nxn) and x is an element of R^K . r is a real variable that is equal to k. Sorry for misstating the problem statement.
     
  5. Sep 28, 2015 #4

    Mark44

    Staff: Mentor

    Is there any relationship between k and n? Such as ##k \le n##?

    If not, are there an arbitrary number of factors in M? In what you wrote, there last component of x in the matrix product is xr. The exact statement of the problem would be helpful.

    I don't know if this is the best approach, but I would start with small matrices, say 2 x 2 or 3 x 3 matrix, to get an idea of how things work. I can't give any more advice until I understand more of the details of the problem.
     
  6. Sep 29, 2015 #5
    A is an nxn matrix, I is its identity and should be also an nxn matrix, k must be equal to n because we want A-xrI . Here is the exact statement of the problem:
    Let A ∈Rnxn, x ∈ Rk. Find the first column of M = (A − x1I)(A − x2I)...(A − xkI) using a sequence of GAXPY’s operations.
     
  7. Sep 29, 2015 #6

    Mark44

    Staff: Mentor

    Try what I suggested at the end of post #4, with a 3 x 3 matrix A. That's what I would start with, and it might give you some insight into what happens for larger matrices.

    $$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\
    a_{21} & a_{22} & a_{23} \\
    a_{31} & a_{32} & a_{33} \end{bmatrix}$$
    $$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$
     
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