Is $f(H)=H^TH$ convex? $H$ is a $m\times n$ matrix I tried to prove $f(H)=H^TH$ convex, where $H$ is a matrix. We know when $h$ is a vector, then $f(h)=h^Th$ is convex. Can I prove it using the following equation? 
$[\theta H_1 + (1-\theta) H_2]^T[\theta H_1 + (1-\theta) H_2] \leq \theta H_1^TH_1 + (1-\theta)H_2^TH_2$
In the convex analysis textbook, the definition of a convex function is 
$f(\theta x + (1-\theta)y) \leq \theta f(x) + (1-\theta) f(y)$
where $f(x):R^n\rightarrow R$. But for my case, $f(H)=H^TH$ is $R^{m\times n} \rightarrow R^{n\times n}$. Please help to prove or disprove it? Thanks

For symmetric matrices $A$ and $B$, the inequality $A \leq B$ means that $B-A$ is positive semi-definitive:
$$((B-A)x, x) \geq 0 \quad \text{for any } x.$$
So we fix $x \in \Bbb{R}^{n}$ and check if
$$ (f(\theta H_{1} + (1-\theta)H_{2})x, x) \leq \theta ( f(H_{1}) x, x) + (1-\theta) (f(H_{2})x, x)$$
holds. That is, the convexity of $f$ is equivalent to the convexity of $H \mapsto (f(H)x, x)$ for any $x$. But since $(f(H)x, x) = (H^{T}Hx, x) = \| Hx \|^{2}$, it follows that
\begin{align*}
(f(\theta H_{1} + (1-\theta)H_{2})x, x)
&= \| \theta H_{1}x + (1-\theta)H_{2}x \|^{2} \\
{\scriptsize (\because \text{ triangle inequality})} \quad &\leq \left( \| \theta H_{1}x \| + \|(1-\theta)H_{2}x \| \right)^{2} \\
&= \left( \theta \| H_{1}x \| + (1-\theta) \|H_{2}x \| \right)^{2} \\
{\scriptsize (\because \text{ convexity of } t \mapsto t^{2})} \quad &\leq \theta \|H_{1}x\|^{2} + (1-\theta) \|H_{2}x\|^{2} \\
&= \theta (f(H_{1})x, x) + (1-\theta) (f(H_{2})x, x).
\end{align*}
This proves the convexity of $f$ as desired.
