# Thread: 2x2 Positive Definite matrix.

1. ## 2x2 Positive Definite matrix.

$\displaystyle A = \left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right]$

Show that if A is positive definite, then both a & b are positive.

I think I'm on the right track with the following...

$\displaystyle z^TAz = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right]\left[ \begin{array}{cccc} Z_1 \\ Z_2 \end{array} \right]$

$\displaystyle = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} aZ_1 + cZ_2 \\ cZ_1 + bZ_2 \end{array} \right]$

$\displaystyle = Z_1(aZ_1 + cZ_2) + Z_2(cZ_1 + bZ_2)$

$\displaystyle = aZ_1^2 + 2CZ_1Z_2 + bZ_2^2$

$\displaystyle aZ_1^2 + 2CZ_1Z_2 + bZ_2^2 > 0$

Can anyone point me in the right direction? Thanks.

2. Originally Posted by sean.1986
$\displaystyle A = \left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right]$

Show that if A is positive definite, then both a & b are positive.

I think I'm on the right track with the following...

$\displaystyle z^TAz = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} a & c \\ c & b \end{array} \right]\left[ \begin{array}{cccc} Z_1 \\ Z_2 \end{array} \right]$

$\displaystyle = \left[ \begin{array}{cccc} Z_1 & Z_2 \end{array} \right]\left[ \begin{array}{cccc} aZ_1 + cZ_2 \\ cZ_1 + bZ_2 \end{array} \right]$

$\displaystyle = Z_1(aZ_1 + cZ_2) + Z_2(cZ_1 + bZ_2)$

$\displaystyle = aZ_1^2 + 2CZ_1Z_2 + bZ_2^2$

$\displaystyle aZ_1^2 + 2CZ_1Z_2 + bZ_2^2 > 0$

Can anyone point me in the right direction? Thanks.
your final result must hold for all real numbers $\displaystyle Z_1,Z_2.$ what do you get if you put $\displaystyle Z_1=1, \ Z_2=0$ or $\displaystyle Z_1=0, \ Z_2=1$?

3. Originally Posted by NonCommAlg
your final result must hold for all real numbers $\displaystyle Z_1,Z_2.$ what do you get if you put $\displaystyle Z_1=1, \ Z_2=0$ or $\displaystyle Z_1=0, \ Z_2=1$?
I see. So the incompatibility with those values would stop it from being positive definite. I think I was looking at it from the wrong perspective. Rather than trying to find some z for which a and b wouldn't work unless they were both positive, I was looking more generally at a and b to make sure they always had to be positive for the inequality to hold true.

Thanks for the help.