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國立政治大學 2025 · 考試 · 第 14 題

Which one is false for the following statements:

(A)
If A is diagonalizable, then there is a basis for eigenvectors of A
(B)
Not all $2 \times 2$ matrices are diagonalizable
(C)
If A does not have n distinct eigenvalues, then A is not diagonalizable
(D)
Let $A = \begin{bmatrix}1 & 2 & 3 & 4\\ 2 & 4 & 1 & 3\\ 4 & 2 & 3 & 1\\ 3 & 1 & 4 & 2\end{bmatrix}$ , then $[1,1,1,1]^T$ is an eigenvector of $A$ .
(E)
Eigenvectors corresponding to different eigenvalues are linearly independent.

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