# Brockett cost function on the Stiefel manifold

In section 4.8 of Optimization Algorithms on Matrix Manifolds (3rd version), Absil, Mahony & Sepulchre define the cost function $$f : \mbox{St} (p,n) \to \mathbb{R}, \qquad X \mapsto \mbox{tr} \left( X^{T} A X N \right)$$ with $$N = \mbox{diag} \left(\mu_1, \ldots, \mu_p\right)$$ with $$0 \leq \mu_1 \leq \ldots \leq \mu_p$$.

I would like to play around with this cost function term with explicit matrices. However, I am a bit unsure about this $$N$$ matrix. How does it look like and what are the $$\mu_i$$'s?

Before introducing this matrix cost function expression they say that "we now consider a cost function defined as a weighted sum" $$\sum_{i} \mu_i x^{T}_{(i)} A x_{(i)}$$ of Rayleight quotients on the sphere under and orthogonality constraint, $$x^{T}_{i} x_{j}= \delta_{ij}$$.

• Read carefully the passages before the definition. There must be a description of $A$ and $N$.
– Laz
Feb 24 at 12:26
• @Laz, thank you for your reply that was extremely useful. Feb 24 at 12:36
• If you take the square root of $N$, notice that one is weighting the columns of $X$ using the weights $\sqrt{\mu_i}$ Feb 24 at 18:41
• Do you agree with my edits? Feb 24 at 18:41

Define $$\mathbf{y}_n = \mu_n \mathbf{x}_n$$, $$\mathbf{X}= \begin{bmatrix} \mathbf{x}_1 & \ldots & \mathbf{x}_N \end{bmatrix}$$ and similarly for $$\mathbf{Y} = \begin{bmatrix} \mathbf{y}_1 & \ldots & \mathbf{y}_N \end{bmatrix} =\mathbf{XN}$$
The cost function writes $$\phi =\sum_n \mathbf{x}_n^T \mathbf{A} \mathbf{y}_n =\mathrm{tr} \left( \mathbf{X}^T\mathbf{A}\mathbf{Y} \right) =\mathrm{tr} \left( \mathbf{X}^T\mathbf{A}\mathbf{XN} \right)$$