is a positive definite matrix necessarily symmetric?

Also if A is an nxn matrix and is singular (not invertible) then there is an x in Rn such that Ax=0 right?

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- June 7th 2008, 12:11 PMmatt beesonlinear algebra quick question
is a positive definite matrix necessarily symmetric?

Also if A is an nxn matrix and is singular (not invertible) then there is an x in Rn such that Ax=0 right? - June 7th 2008, 07:00 PMNonCommAlg
No! for example the matrix is positive definite but not symmetric. to see why is positive

definite, let . then see that

Quote:

Also if A is an nxn matrix and is singular (not invertible) then there is an x in Rn such that Ax=0 right?

- June 7th 2008, 07:18 PMtopsquark
- June 7th 2008, 09:07 PMNonCommAlg
No! being Hermitian is not a part of the definition. an matrix over is positive definite if

if all entries of are real, then it's easy to see that in order to prove that is positive definite, we only need to show that

using this fact, we can see easier why the matrix in my previous post is positive definite.

if is Hermitian, then is positive definite if because for a Hermitian matrix - June 7th 2008, 09:23 PMmr fantastic
- June 7th 2008, 09:53 PMNonCommAlg
- June 7th 2008, 11:36 PMmr fantastic