The Procrustes Problem

The Procrustes problem (see [158]) is the minimization of $\vert\vert AY-YB\vert\vert _F$ for constant $A$ and $B$ over the manifold $Y^*Y=I$. This minimization determines the nearest matrix $\hat{A}$ to $A$ for which

$$Q^* \hat{A} Q = \left[ \begin{array}{cc} B & * \cr 0 & * \end{array} \right];$$

i.e. the columns of $B$ span an invariant subspace of $\hat{A}$.

The differential of $F(Y) = \frac{1}{2}\vert\vert AY - YB\vert\vert _F^2 = \frac{1}{2} \tr(AY-YB)^*(AY-YB)$ is given by

$$dF(Y) = A^*(AY-YB)-(AY-YB)B^*.$$

This can be derived following the process outlined above. Observe that

$$\begin{eqnarray*} \frac{d}{dt} F(Y(t)) \vert _{t=0}&=&\frac{1}{2}\tr((AV-VB)^*(A... ...tr((AV-VB)^*(AY-YB)) \\ &=& \tr(V^*(A^*(AY-YB))-V^*(AY-YB)B^*). \end{eqnarray*}$$

The second derivative of $F(Y)$ is given by the equation

$$\frac{d}{dt} dF(Y(t))\vert _{t=0} = A^*(AH-HB)-(AH-HB)B^*,$$

where $\dot{Y}(0) = H$, which can be obtained by varying the expression for $dF$.