1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Heeeeeeelp pleeeeeease

  1. Oct 4, 2007 #1
    Hey,

    Could someone help me with my question here please ??

    it's a linear algebra que ....

    let the (matrix) A= a b
    c d

    Find a polynomial p(x) = x^2 + qx + r
    such that p(A)=0
     
  2. jcsd
  3. Oct 4, 2007 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Look up and learn about characteristic polynomials. This question didn't come out of a vacuum. There's a chapter in your book about it.
     
  4. Oct 4, 2007 #3
    unfortunately not !!!!

    nothing in my book says anything about polynomial .. i searched all the day for that !
     
  5. Oct 4, 2007 #4

    dynamicsolo

    User Avatar
    Homework Helper

    Although it looks strange, you can set the variable in a polynomial equal to a matrix. What would x^2 mean if x = A? The next term would be qA , which has a clear meaning. The last term has to be interpreted in terms of matrices: in numerical algebra, r = r·1 , so in matrix algebra, r = r · I . You can now add the terms of your polynomial, obeying matrix addition. You are looking to express q and r in terms of the entries in A, so that the sum is 0, that is, the zero matrix.
     
  6. Oct 4, 2007 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Try eigenvalues. The quick story is that A satisfies the polynomial determinant(A-x*I)=0 where I is the identity matrix and x is the variable in your polynomial. Does that sound familiar?
     
  7. Oct 5, 2007 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    A linear algebra textbook that doesn't say anything about characteristic polynomials? I find that hard to believe. In fact, from the nature of this problem, I suspect they may be introduced in the very next section: this problem looks like an introduction to them!

    But since you have not yet seen characteristic polynomials (and have already spent a day on this), how about the obvious: Go ahead and calculate x2, for x equal to this matrix, put it into the equation x2+ qx+ rI (that last term is the constant r multiplying the identity matrix in order to make it a matrix equation) and see what p and q must be in order that it equal the 0 matrix.

    You will get four equations for the two variables, p and r, but they are not all independent.
     
    Last edited: Oct 5, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?