1. ## Positive Definite Proof

We say a symmetric matrix A is positive definite if Ax dot x>0 for all x not zero, negative definite if Ax dot x<0 for all x not zero and positive semi-definite if Ax dot x >/= 0 for all x.

Show that if A and B are positive definite, then so is A+B

Show that if A is positive definite if and only if all its eigenvalues are positive.

2. ## Re: Positive Definite Proof

For the first question, write $\displaystyle (A+B)x\cdot x$ in function of $\displaystyle Ax\cdot x$ and $\displaystyle Bx\cdot x$. For the second question, take $\displaystyle x$ an eigenvector for the eigenvalue $\displaystyle \lambda$.

3. ## Re: Positive Definite Proof

Alright, i figured out the first part, but I'm a little unsure how you see to proceed for b?

4. ## Re: Positive Definite Proof

Let $\displaystyle \lambda$ an eigenvalue for $\displaystyle A$, and $\displaystyle x\neq 0$ an eigenvector. Then $\displaystyle Ax=\lambda x$ and $\displaystyle x^tAx=\lambda \lVert x\rVert^2\geq 0$ so $\displaystyle \lambda \geq 0$. Now, to see the converse, use spectral theorem.