Thread: positive definite matrix

Let A be the following symmetric matrix
A =
[4 2 -2 2
2 2 1 1
-2 1 14 -1
2 1 -1 x]

find the values of x , such that A is positive definite.

a symmetric matrix is a square matrix, A, that is equal to its transpose and An n × n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (i.e. z ∈ Rn), where zT denotes the transpose of z.

Originally Posted by tinng

Let A be the following symmetric matrix
A =
[4 2 -2 2
2 2 1 1
-2 1 14 -1
2 1 -1 x]

find the values of x , such that A is positive definite.

There is a theorem telling that (here, ) is positive definite if, and only if, for the determinant of is positive.

If you know this, then it is easy. The determinants of are positive, they do not depend on , and the determinant of is found to be , hence it is positive iff . So is positive definite iff .