# A FEAST Algorithm for the Linear Response Eigenvalue Problem
## 2 Preliminaries  
\(\mathbb{R}^{m\times n}\) is the set of all \(m\times n\) real matrices. For \(\mathcal{X}\subseteq\mathbb{R}^{n}\), \(\dim(\mathcal{X})\) is the dimension of \(\mathcal{X}\). For \(X\in\mathbb{R}^{m\times n}\), \(X^{\mathrm{T}}\) is its transpose and \(\mathcal{R}(X)\) the column space of \(X\), and the submatrix \(X_{\langle ij\rangle}\) of \(X\) consists of column \(i\) to column \(j\). \(I_{n}\) is the \(n\times n\) identity matrix or simply \(I\) if its dimension is clear from the context. For matrices or scalars \(X_{i}\), \(\operatorname{diag}(X_{1},X_{2},\ldots,X_{k})\) denotes the block diagonal matrix with diagonal block \(X_{1},X_{2},\ldots,X_{k}\). For any given symmetric and positive definite matrix \(W\in\mathbb{R}^{N\times N}\), the \(W\)-inner product and its induced \(W\)-norm are defined by:  
\[\langle x,y\rangle_{w}=y^{\mathrm{T}}Wx,\quad\|x\|_{w}=\sqrt{\langle x,x \rangle_{w}}.\]  
If \(\langle x,y\rangle_{w}=0\), then we say \(x\perp_{w}y\) or \(y\perp_{w}x\). The projector \(\Pi_{w}\) is said to the \(W\)-orthogonal projector onto \(\mathcal{X}\) if for any vector \(x\in\mathbb{R}^{N}\),  
\[\Pi_{w}\,x\in\mathcal{X}\quad\text{and}\quad(I-\Pi_{w})x\perp_{w}\mathcal{X}.\]  
For two subspaces \(\mathcal{X},\,\mathcal{Y}\subseteq\mathbb{R}^{N}\) with \(k=\dim(\mathcal{X})\leq\dim(\mathcal{Y})=\ell\), let \(X\) and \(Y\) be the \(W\)-orthonormal basis matrices of subspaces \(\mathcal{X}\) and \(\mathcal{Y}\), respectively, i.e.,  
\[X^{\mathrm{T}}WX=I_{k},\ \mathcal{R}(X)=\mathcal{X},\quad\text{and}\quad Y^{ \mathrm{T}}WY=I_{\ell},\ \mathcal{R}(Y)=\mathcal{Y}.\]  
Denote the singular values of \(X^{\mathrm{T}}WY\) by \(\sigma_{j}\) for \(1\leq j\leq k\) in ascending order, i.e., \(\sigma_{1}\leq\cdots\leq\sigma_{k}\). The diagonal matrix of \(W\)-canonical angles from (If \(\ell=k\), we may say that these angles are between \(\mathcal{X}\) and \(\mathcal{Y}\)[25].) \(\mathcal{X}\) to \(\mathcal{Y}\) in descending order is defined by:  
\[\Theta_{w}(\mathcal{X},\mathcal{Y})=\operatorname{diag}(\theta_{w}^{\alpha_{ 1}}(\mathcal{X},\mathcal{Y}),\ldots,\theta_{w}^{\alpha_{k}}(\mathcal{X}, \mathcal{Y})),\]  
where \(\theta_{w}^{\alpha_{1}}(\mathcal{X},\mathcal{Y})\geq\cdots\geq\theta_{w}^{ \alpha_{2}}(\mathcal{X},\mathcal{Y})\) and:  
\[0\leq\theta_{w}^{\alpha_{1}}(\mathcal{X},\mathcal{Y})=\arccos\sigma_{j}\leq \frac{\pi}{2},\quad\text{for}\,1\leq j\leq k.\]  
In what follows, we sometimes place a vector or matrix in one or both arguments of \(\Theta_{w}(.,. )\) with the understanding that it is about the subspace spanned by the vector or the columns of the matrix argument. Several theoretical properties of the canonical angle and LREP have been established in [25] and [3], respectively. In particular, the following lemmas are necessary for our later developments. **Lemma 1** ([9] Lemma 3.2). : _Let \(\mathcal{X}\) and \(\mathcal{Y}\) be two subspaces in \(\mathbb{R}^{N}\) with equal dimension \(\dim(\mathcal{X})=\dim(\mathcal{Y})=k\). Suppose \(\theta_{w}^{\alpha_{1}}(\mathcal{X},\mathcal{Y})<\pi/2\). Then, for any set \(\{y_{1},y_{2},\ldots,y_{k_{1}}\}\) of the basis vectors in \(\mathcal{Y}\) where \(1\leq k_{1}\leq k\), there is a set \(\{x_{1},x_{2},\ldots,x_{k_{1}}\}\) of linear independent vectors in \(\mathcal{X}\) such that \(\Pi_{w}x_{j}=y_{j}\) for \(1\leq j\leq k_{1}\), where \(\Pi_{w}\) is the \(W\)-orthogonal projector onto \(\mathcal{Y}\)._  
**Lemma 2** ([3] Theorem 2.3). : _The following statements hold for any symmetric matrices \(K\), \(M\in\mathbb{R}^{N\times N}\) with \(M\) being positive definite._  
1. _There exists a nonsingular_ \(\Psi\in\mathbb{R}^{N\times N}\) _such that:_ \[K=\Psi\Lambda^{2}\Psi^{\mathrm{T}}\quad\text{and}\quad M=\Phi\Phi^{\mathrm{T}},\] _where_ \(\Lambda=\mathrm{diag}(\lambda_{1},\lambda_{2},\ldots,\lambda_{N})\)_,_ \(\lambda_{1}^{2}\leq\lambda_{2}^{2}\leq\cdots\leq\lambda_{N}^{2}\) _and_ \(\Phi=\Psi^{-\mathrm{T}}\)_._
2. _If_ \(K\) _is also definite, then all_ \(\lambda_{i}>0\)_, and_ \(H\) _is diagonalizable:_ \[H