Be $\mathcal{W}=\{W=(w_1|\dots|w_p)\in\mathbb{R}^{n\times p}\;:\;W^TW = I\}$ the set of $p$-dimensional orthonormal frames in $\mathbb{R}^n$. Consider $$ L(W) = \mathrm{tr}(MP_W) = \mathrm{tr}(MWW^T) = \sum_{i=1}^pw_i^TMw_i, $$ where $M\in\mathbb{R}^{n\times n}$ is a symmetric positive semi-definite matrix. This means that we can write $M=\sum_{i=1}^n\lambda_iv_iv_i^T$, where $\lambda_1\geq\dots\geq\lambda_n\geq0$ and $v_1,\dots,v_N$ is an orthonormal basis of $\mathbb{R}^n$; also, if $V = (v_1|\dots|v_p)$ then $$L(W)\leq L(V) \qquad \text{for every } W \in \mathcal{W}.$$ My question is: Given $W\in\mathcal{W}$, how can I construct $W_t\in\mathcal{W}$, $t\in[0,1]$, continuous such that $W_0=W$, $W_1=V$ and $L(W_t)\geq L(W)$?


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