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mathshelpstudent8

Member

Registered: 2008-05-13

Posts: 1

Positive definite matrix test

Matrix A=
5k 1
9 1

calculate (transpose(x)*A*x ) where x=
x1
x2

Hence
find the set of values for k where A is positive definite using your result of (transpose(x)*A*x ) for all non zero x in R^2

Last edited by mathshelpstudent8 (2008-05-13 21:23:59)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Positive definite matrix test

Multiplying 3 matrices is just multiplying 2 matrices twice.

Start with, say, Ax.

Ax = (5k  1) (x1)  =  (5kx1 + x2)
        ( 9   1) (x2)      (9x1   + x2)

Now you have Ax as a vector, so you can work out x*(Ax), a scalar.

A matrix is defined to be positive definite if x*Ax > 0, for any non-zero real vector x.
Therefore, once you have your scalar, you need to find what value of k will guarantee it to be positive, regardless of what x1 and x2 are (excluding when they're both 0).

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It wanted to be normal.