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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, $x^{T}_{i} x_{j}= \delta_{ij}$.

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

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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 $$ \phi =\sum_n \mathbf{x}_n^T \mathbf{A} \mathbf{y}_n =\mathrm{tr} \left( \mathbf{X}^T\mathbf{A}\mathbf{Y} \right) =\mathrm{tr} \left( \mathbf{X}^T\mathbf{A}\mathbf{XN} \right) $$

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