What is the classification of this optimization problem:
Find $X$ that maximizes: $$ \text{Find X that maximizes:} \,\,\, \sum_{i = 1}^n y_i^TXX^Ty_i \\ S.T. \,\,\,X^TX = I $$
I read that the optimal solution is related to the eigenvectors of the matrix $\sum y_i y_i^T$. I have no idea why, so I want to find some reference. That's why I need the classification.
I know from Mathematica that where $X$ is a $3 \times 3$ matrix, $$ X = \left( \begin{array}{ccc} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \\ \end{array} \right) $$
Then, $$y_1^TXX^Ty_1 = y_1 \left(x_{11} \left(x_{11} y_1+x_{21} y_2+x_{31} y_3\right)+x_{12} \left(x_{12} y_1+x_{22} y_2+x_{32} y_3\right)+x_{13} \left(x_{13} y_1+x_{23} y_2+x_{33} y_3\right)\right)+y_2 \left(x_{21} \left(x_{11} y_1+x_{21} y_2+x_{31} y_3\right)+x_{22} \left(x_{12} y_1+x_{22} y_2+x_{32} y_3\right)+x_{23} \left(x_{13} y_1+x_{23} y_2+x_{33} y_3\right)\right)+y_3 \left(x_{31} \left(x_{11} y_1+x_{21} y_2+x_{31} y_3\right)+x_{32} \left(x_{12} y_1+x_{22} y_2+x_{32} y_3\right)+x_{33} \left(x_{13} y_1+x_{23} y_2+x_{33} y_3\right)\right) $$
So, doesn't look convex to me, but I realize there are many tricks. I guess it is polynomial optimization?
The soution was referenced to a textbook: Bellman, R., 1970. Introduction to Matrix Analysis, 2nd Ed,McGraw-Hill, New York. but I can't find the problem in the textbook. The text does have a chapter on "dynamic programming", but I couldn't find this problem. Would it be a dynamic programming problem?