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Math Help - Matrices (transpose with unknowns)

  1. #1
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    Matrices (transpose with unknowns)

    Matrix A=
    5k 1
    9 1

    calculate (transpose(x)*A*x ) where x=
    x1
    x2
    find the set of values for k where (transpose(x)*A*x ) > 0 for all non zero x in R^2

    I am having real difficulty on this one, as when i did this calculation i got:

    5kx1^2 + 10x1x2 + x2^2

    Now it says for non-zero x but that means they can still be negative? Please help.
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  2. #2
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    Quote Originally Posted by mathsstudent8 View Post
    Matrix A=
    5k 1
    9 1

    calculate (transpose(x)*A*x ) where x=
    x1
    x2
    find the set of values for k where (transpose(x)*A*x ) > 0 for all non zero x in R^2

    I am having real difficulty on this one, as when i did this calculation i got:

    5kx1^2 + 10x1x2 + x2^2

    Now it says for non-zero x but that means they can still be negative? Please help.
    I don't see what the confusion is ...... You require the values of k such that 5 k x_1^2 + 10 x_1 x_2 + x_2^2 > 0 for all non-zero values of x_1 and x_2.

    In other words, k > -\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right) for all non-zero values of x_1 and x_2.

    It's not hard to see that the least upper bound of -\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right) is zero. Therefore .....

    Alternatively, you could think of 5 k x_1^2 + 10 x_1 x_2 + x_2^2 = 0 as representing a conic section ......
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