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國立政治大學 2026 · 考試 · 第 8 題

考科線性代數

For problems 8-10, please find a singular value decomposition for the following matrix:

B=[3000−3101−3]=UΣVTB = \begin{bmatrix}3 & 0 & 0\\ 0 & -3 & 1\\ 0 & 1 & -3\end{bmatrix} = U\Sigma V^T

(4分) U=?U = ?

  • (A)
    [010120−12−120−12]\begin{bmatrix}0 & 1 & 0\\ \frac{1}{\sqrt2} & 0 & -\frac{1}{\sqrt2}\\ -\frac{1}{\sqrt2} & 0 & -\frac{1}{\sqrt2}\end{bmatrix}
  • (B)
    [10001212012−12]\begin{bmatrix}1 & 0 & 0\\ 0 & \frac{1}{\sqrt2} & \frac{1}{\sqrt2}\\ 0 & \frac{1}{\sqrt2} & -\frac{1}{\sqrt2}\end{bmatrix}
  • (C)
    [010−1201212012]\begin{bmatrix}0 & 1 & 0\\ -\frac{1}{\sqrt2} & 0 & \frac{1}{\sqrt2}\\ \frac{1}{\sqrt2} & 0 & \frac{1}{\sqrt2}\end{bmatrix}
  • (D)
    [13121613−1216130−26]\begin{bmatrix}\frac{1}{\sqrt3} & \frac{1}{\sqrt2} & \frac{1}{\sqrt6}\\ \frac{1}{\sqrt3} & -\frac{1}{\sqrt2} & \frac{1}{\sqrt6}\\ \frac{1}{\sqrt3} & 0 & -\frac{2}{\sqrt6}\end{bmatrix}

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