Let $M$ be a square $n\times d$ matrix (i.e., $n\times d$ row and $n\times d$ column). Let $\xi^{1},\,.\,.\,.\,,\xi^{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have $\xi^{i}=(\lambda_1,0,\,.\,.\,.\,,0)+.\,.\,.\,+(0,\,.\,.\,.\,,0,\lambda_n)$, where $0\in \mathbb{R}^{d}$ and $\lambda_j\in \mathbb{R}^{d}$. We can write $\xi^{i}$ as $$\xi^{i}=\alpha_{i,1}(\tilde{\lambda}_1,0,\,.\,.\,.\,,0)+\alpha_{i,2}(0,\tilde{\lambda}_2,0,\,.\,.\,.\,,0)+.\,.\,.\,+\sqrt{1-\alpha_{i,1}^2-\,.\,.\,.\,-\alpha_{i,n-1}^2}(0,\,.\,.\,.\,,0,\tilde{\lambda}_n).$$ we want to prove that there exist $\alpha_0>0$ small enough and $i_1,\,.\,.\,.\,,i_{(n-1)\times d}$ such that $$\alpha_{i_k,l}>\alpha_0\qquad\text{for}\;k=1,\,.\,.\,.\,,(n-1)\times d\;\text{and}\; l=1,\,.\,.\,.\,n-1.$$
Eigenvectors of a matrix
yassine yassine
- 41
- 5