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Show matrix is symmetric and positive definite watch

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    I got a question as follows,

    Let A be a non-singular real square matrix. Show that (A^T)A is symmetric and postive definite.

    I know how to spot when a matrix is symmetric and also how to prove that it is positive definite, but im not quite sure how to start this question as it is for a general nxn matrix.

    Does anyone know how to show this? or could maybe point me in a general direction of how to start.
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    So it is symmetric if B^T = B. Can you check this for the expression you have?
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    (Original post by King Ripper)
    I got a question as follows,

    Let A be a non-singular real square matrix. Show that (A^T)A is symmetric and postive definite.

    I know how to spot when a matrix is symmetric and also how to prove that it is positive definite, but im not quite sure how to start this question as it is for a general nxn matrix.

    Does anyone know how to show this? or could maybe point me in a general direction of how to start.
    Like Simon says, let B = A^T * A; you need to check that B^T = B. Do you know how to transpose a product of matrices?

    A symmetric matrix B is positive definite if xTBx > 0 for all vectors x =/= 0. Can you write this in terms of a statement about Ax? (Note that yTz = y . z (dot product) when y and z are real vectors.)
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    So if i have,

    B = A^T * A

    B^T = (A^T * A)^T

    B^T = (A^T)A

    B^T = B

    so that shows it is symmetric right?

    Im still not sure how you would show it is positive definite by using (x^T)Bx > 0 for all vectors x =/= 0
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    (Original post by King Ripper)
    So if i have,

    B = A^T * A

    B^T = (A^T * A)^T

    B^T = (A^T)A

    B^T = B

    so that shows it is symmetric right?
    Yep.

    (Original post by King Ripper)
    Im still not sure how you would show it is positive definite by using (x^T)Bx > 0 for all vectors x =/= 0
    Write B = A^T * B again, and try and manipulate it into a statement about the vector (Ax). (Hint: what's (Ax)^T?)
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    Thanks for all the help so far,

    (Ax)^T = (x^T)(A^T)

    am i right in starting off by multiplying both side by the vector x to give me,

    Bx = ((A^T)(A))x

    Bx = (A^T)x Ax

    Im not really sure to go from there, assuming im going in the right direction to start with?

    From your hint i tried to take the transpose of both sides but that just made things worse i think
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    I'd go the other way. Let B be any old matrix. How would you write the condition that B is positive definite? (i.e. what's the definition of a positive definite matrix).

    Now look at what this looks like in the case B = A^T A.
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    (Original post by King Ripper)
    Bx = ((A^T)(A))x

    Bx = (A^T)x Ax
    What you've done here is wrong. And I'm not sure why you're not looking at xTBx...
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    Thanks for the help everyone, I finally got it . Took me a while though.

    I didnt realise earlier that you meant to write,

    x^T((A^T) A) x

    then i can see that it is just the vector Ax dotted with itself which is of course positive.

    Thanks again for the help.
 
 
 
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