Positive definite forms

Dear Sir,

This is a question regarding this week’s Algebra 3: Sheet 8.

It’s evidently clear that a function is positive definite if (x^t)Ax > 0.

So for example on question 1(a), is it ok to say the matrix is positive definite if all its principle minors have positive determinants:

here is my working so far for 1(a): let the matrix be A.

———————————
det(A_1) = 1
det(A_2) = (1×3) – (2×2) = -1.

Since the determinant for the 2nd principle minor is negative, then it cannot be positive definite? is this the correct way of solving the problem?

thanks.
kind regards šŸ™‚

Reply:

Within the context of this problem, I would say no, that is not a good way. You are referring to the principal minors test for positive definiteness of a symmetric matrix. However, since we have not proved it in class, you shouldn’t use it at this point unless you are prepared to prove it yourself. The easiest way to approach this problem would be to find the diagonal representative using double operations for (a) and (b). For (c), (d), (e) you should be able to figure it out by thinking of the definition carefully.

MK

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