How do we know we're maximizing the Lagrangian objective function in PCA?

In Principal Component Analysis, we start with $$m$$ observations $$x_1,\dots,x_m$$, each of which is an $$n$$-dimensional vector. Assume we have centered the data; that is, we have subtracted the variable means from each observation. To project each observation into a $$1$$-dimensional subspace while maximizing sample variance, we want to compute a unit length basis vector, call it $$v$$. So we get the constrained problem

$$\mathop{\arg\,\max}\limits_{v}\,\frac{1}{m}\sum_{i=1}^{m}\left\lVert\langle x_i,v\rangle v\right\rVert$$

$$\text{subject to }v^Tv=1$$

After some computations, this becomes

$$\mathop{\arg\,\max}\limits_{v}\,\frac{1}{m}\sum_{i=1}^{m}v^TCv$$

$$\text{subject to }v^Tv=1$$

where $$C$$ is the covariance matrix of the data. Then the Lagrangian objective function is

$$\mathcal{L}(v,\lambda)=v^TCv-\lambda(v^Tv-1)$$

If we start taking partial derivatives of $$\mathcal{L}$$, setting these to zero, etc, how do we know that we are maximizing $$\mathcal{L}$$? Do we need to check that the Hessian is negative definite? Even then, I think that only guarantees a local maximum?

• Because the constraint $v^\prime v=1$ defines a compact domain and the objective function is continuously differentiable, you are guaranteed that at least one critical point is a global maximum. – whuber Dec 4 '18 at 19:01
• @whuber And not a global minimum? – waynemystir Dec 4 '18 at 19:17
• It seems the relevant answers on this site don't attempt to check all the critical points of $\mathcal{L}$ (for example, stats.stackexchange.com/questions/10251/… stats.stackexchange.com/questions/301532/…) – Frank Dec 4 '18 at 19:51
• Thanks Frank, good point and very helpful links. I'll also add this stats.stackexchange.com/questions/217995/… where @amoeba 's first proof uses Lagrange multipliers. Aside from checking critical points, I would think a negative definite Hessian would suffice. – waynemystir Dec 4 '18 at 20:38
• Whether you need to check for a global minimum versus a global maximum depends on the algorithm. Many of them simply iteratively apply the matrix, relying on the fact that the result has probability $1$ of converging exponentially to the maximum rather than the minimum. – whuber Dec 5 '18 at 19:16