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文中把k-ellipse的方程表示成行列式，要用到以下的Lemma：
Lemma 2.2. Let $M_1, \ldots, M_k$ be symmetric matrices, let $U_1, \ldots, U_k$ be orthogonal matrices, and let $\Lambda_1, \ldots, \Lambda_k$ be diagonal matrices such that $M_i=U_i \cdot \Lambda_i \cdot U_i^T$ for $i=1, \ldots, k$. Then
$$
\left(U_1 \otimes \cdots \otimes U_k\right)^T \cdot\left(M_1 \oplus \cdots \oplus M_k\right) \cdot\left(U_1 \otimes \cdots \otimes U_k\right)=\Lambda_1 \oplus \cdots \oplus \Lambda_k
$$
In particular, the eigenvalues of the tensor sum $M_1 \oplus M_2 \oplus \cdots \oplus M_k$ are the sums $\lambda_1+\lambda_2+\cdots+\lambda_k$ where $\lambda_1$ is any eigenvalue of $M_1, \lambda_2$ is any eigenvalue of $M_2$, etc.
但没有给出证明。如何证明呢 |
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