# Math Help - Prove matrx is positive definite

1. ## Prove matrx is positive definite

Show that matrix A =
|a b|
|0 c|

is positive definite if a > 0, c > 0 and |b| < 2(ac)^.5

not really sure how to prove this, i multiplied an vector z = [x,y]^T so that

z^T * A * z = ax^2 + byx + cy^2 > 0 but i am not sure is this is even the right direction.

2. Originally Posted by beachbump
Show that matrix A =
|a b|
|0 c|

is positive definite if a > 0, c > 0 and |b| < 2(ac)^.5

not really sure how to prove this, i multiplied an vector z = [x,y]^T so that

z^T * A * z = ax^2 + byx + cy^2 > 0 but i am not sure is this is even the right direction.

The expression $ax^2+byx+cy^2$ is a quadratic in $x$ with positive quadratic coefficient and thus it will ALWAYS be positive iff its discriminant is negative:

$\Delta = b^2y^2-4acy^2<0\Longleftrightarrow b^2-4ac<0\Longleftrightarrow |b|<2\sqrt{ac}$

Tonio