TITLE: If $A,B$ symmetric positive semidefinite, show tr$(AB) \geq 0$
QUESTION [10 upvotes]: Supposing $V$ is a finite dimensional vector space (over $\mathbb{R}$) of dimension $n$, and $A,B$ are symmetric positive definite linear mappings from $V$ to $V$, how can I show that in any orthonormal basis $\mathrm{tr}(AB) \geq 0$?
I noticed that since they are symmetric we have that 
$$\mathrm{tr}(AB) = \sum_{i=1}^n\sum_{j=1}^nA_{ij}B_{ji} = \sum_{i=1}^n\sum_{j=1}^nA_{ij}B_{ij}$$ which is the sum of the elements of the element-wise product of $A,B$. I don't know if this is helpful.

REPLY [4 votes]: Here's another derivation (7 years later):
Let $A,B\succeq0$. Then the eigendecomposition of symmetric $B$ gives $B=\sum_{i=1}^n \lambda_i v_i v_i^T$. Therefore,
$$\begin{align}
\operatorname{Tr}[AB]&=\operatorname{Tr}[A\sum_{i=1}^n \lambda_i v_i v_i^T]\\
&=\sum_{i=1}^n \lambda_i \operatorname{Tr}[Av_i v_i^T]\\
&=\sum_{i=1}^n \underbrace{\lambda_i}_{\geq0} \underbrace{v_i^TAv_i}_{\geq0} \\
&\geq 0
\end{align}$$
where the last equality is from the cyclic property of the trace.
Feel free to ask for any clarifications neeeded.
Edit: Here's the explanation of the eigendecomposition. 
In matrix form, the eigen-equation is: $BV=V\Lambda$, where $V$ is a matrix whose columns are the eigenvectors $\{v_i\}$ of $B$, and where $\Lambda=\operatorname{diag}(\lambda_1,...,\lambda_n)$ is a diagonal matrix with the eigenvalues of $B$ along the diagonal. Because $B$ is symmetric, these $V$ matrices are orthogonal, meaning their columns are orthonormal, so $VV^T=V^TV=I_n$. We can then re-write the matrix equation: $BVV^T=B=V\Lambda V^T$.
$$\begin{align}
\rightarrow B &=
\begin{bmatrix}
v_1 & \cdots & v_n
\end{bmatrix}
\begin{bmatrix}
\lambda_1 & \cdots & 0\\
\vdots & \ddots & \vdots\\
0 & \cdots & \lambda_n
\end{bmatrix}
\begin{bmatrix}
v_1^T\\
\vdots\\
v_n^T
\end{bmatrix}\\
&=
\begin{bmatrix}
v_1 & \cdots & v_n
\end{bmatrix}
\begin{bmatrix}
\lambda_1v_1^T\\
\vdots\\
\lambda_nv_n^T
\end{bmatrix}\\
&=
\sum_{i=1}^n \lambda_i v_i v_i^T
\end{align}$$
where this is a sum of outer products of the eigenvectors. Note that I wrote my matrices above as vector with elements that are vectors. This shorthand is valid and very convenient, but feel free to write it out and check me!