矩阵积(和)的特征值反例

1.设矩阵 \(A, B\) 都只有正的特征值, 则 \(AB\) 的特征值都为正.(Flase)
反例:
\(A = \begin{bmatrix} 10 & 9\\ -8 & -7\\ \end{bmatrix}\), \(B = \begin{bmatrix} 1 & 0\\ 0 & 2\\ \end{bmatrix}\), \(AB = \begin{bmatrix} 10 & 18\\ -8 & -14\\ \end{bmatrix}\), \(\sigma(A) = \{1, 2\}\), \(\sigma(B) = \{1, 2\}\), \(\sigma(AB) = \{-2, -2\}\).

但 \(AB\) 不会有特征值为 0, 因为

\[0\neq det(A)det(B) = det(AB) = \Pi_{\lambda_i\in \sigma(AB)}\lambda_i \]

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2.设矩阵 \(A, B\) 都只有正的特征值, 则 \(A+B\) 的特征值都为正.(Flase)
反例:
\(A = \begin{bmatrix} 5 & -1\\ 9 & -1\\ \end{bmatrix}\), \(B = \begin{bmatrix} 2 & 1\\ 0 & 1\\ \end{bmatrix}\), \(A+B = \begin{bmatrix} 7 & 0\\ 9 & 0\\ \end{bmatrix}\), \(\sigma(A) = \{2, 2\}\), \(\sigma(B) = \{1, 2\}\), \(\sigma(AB) = \{0, 7\}\).

\(A = \begin{bmatrix} 5 & -1\\ 12 & -2\\ \end{bmatrix}\), \(B = \begin{bmatrix} 2 & 1\\ 0 & 1\\ \end{bmatrix}\), \(A+B = \begin{bmatrix} 7 & 0\\ 12 & -1\\ \end{bmatrix}\), \(\sigma(A) = \{1, 2\}\), \(\sigma(B) = \{1, 2\}\), \(\sigma(AB) = \{-1, 7\}\).