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高代:这个求公共特征值思路难想到!
设 与 均为 阶方阵,若关于未知方阵 的方阵方程 只有零解,证明方阵 与 没有公共特征值。
证明:反证法,假设 与 有公共特征值,设为,则存在非零向量,使得 ,于是得到 ,从而有,,因此是方程 的解,由向量非零得是非零解.
English Edition. Let and be matrix. If the equation has only 0 solution, then and have no common eigenvalues.
Proof: We show that by contradiction. Assume and have common eigenvalues, let it be \lambda. So we get where both and are non-zero vectors. Then we have , and ,. Hence is a solution of the equation where .
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