Simultaneous Diagonalization

Another problem like the simultaneous Schur problem, involving sets of matrices with similar structures, is the problem of finding the best set of eigenvectors for a set of symmetric matrices given that the matrices are known to be simultaneously diagonalizable, but may have significant errors in their entries from noise or measurement error. Instances of this problem arise in the psychometric literature, where it is called an INDSCAL problem [107].

Phrased in terms of a minimization, one has a set of symmetric matrices $A_i$ and wishes to find $Y \in {\cal O}(n)$ that minimizes $F(Y) = \frac{1}{2} \sum_i \vert\vert [Y,A_i] \vert\vert _F^2 = \frac{1}{2} \sum_i \vert\vert Y A_i - A_i Y \vert\vert _F^2$.

We then have

$$dF(Y) = \sum_i [[Y, A_i],A_i^*]$$

and

$$\frac{d}{dt} dF(Y(t))\vert _{t=0} = \sum_i [[H, A_i],A_i^*],$$

where $\dot{Y}(0) = H$.