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Math Help - 2x2 Positive Definite matrix.

  1. #1
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    2x2 Positive Definite matrix.

    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...


    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]

    = \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]

    = Z_1(aZ_1 + cZ_2) + Z_2(cZ_1 + bZ_2)

    = aZ_1^2 + 2CZ_1Z_2 + bZ_2^2

    aZ_1^2 + 2CZ_1Z_2 + bZ_2^2 > 0

    Can anyone point me in the right direction? Thanks.
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  2. #2
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    Quote Originally Posted by sean.1986 View Post
    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...


    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]

    = \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]

    = Z_1(aZ_1 + cZ_2) + Z_2(cZ_1 + bZ_2)

    = aZ_1^2 + 2CZ_1Z_2 + bZ_2^2

    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 Z_1,Z_2. what do you get if you put Z_1=1, \ Z_2=0 or Z_1=0, \ Z_2=1?
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    your final result must hold for all real numbers Z_1,Z_2. what do you get if you put Z_1=1, \ Z_2=0 or 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.
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