Positive definiteness

Question

Let $A$ be a positive definite matrix is it true that also $A+A^T$ is positive definite? If it is true, how to prove it? I try to employ the definition using eigenvalues but i'm not able to proceed.

Thanks in advance.

Answer 1

$A$ is pos. def. so we know that: $x^T Ax>0, \forall x\in\mathbb{R}^n\backslash\{0\}$. Let $B = A + A^T$ then we want to know if:

$$x^TBx>0\quad\forall x\in\mathbb{R}^n\backslash\{0\}.$$

Plug in $B = A + A^T$:

$$x^TBx = x^T(A + A^T)x = x^TAx + x^TA^Tx = x^TAx + (x^TAx)^T = x^TAx + x^TAx>0\quad\forall x\in\mathbb{R}^n\backslash\{0\}.$$

In the last step we used that $x^TAx$ is a scalar (see answer by @0XLR)

Answer 2

Hint: $x^T A^T x$ is a scalar so $x^T A^T x = (x^T A^T x)^T = x^T A x$.