每週問題 May 5, 2014

Posted on 05/05/2014 by ccjou

這是關於正交矩陣的主對角分塊的奇異值問題。

Let Q be an real orthogonal matrix partitioned as

Q=\begin{bmatrix} A&B\\ C&D \end{bmatrix},

where A, B, C, D are n\times n. Show that A and D have the same singular values.

 
參考解答：

正交矩陣 Q 滿足 Q^TQ=QQ^T=I，即有

\displaystyle A^TA+C^TC=I_n,~~CC^T+DD^T=I_n。

因此，

\displaystyle A^TA=I_n-C^TC,~~DD^T=I_n-CC^T。

然而，C^TC 和 CC^T 有相同的特徵值，推論 I_n-C^TC 和 I_n-CC^T 也有相同的特徵值，也就是說，A^TA 和 DD^T 有相同的特徵值，故 A 和 D 有相同的奇異值。

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8 Responses to 每週問題 May 5, 2014

  1. Shenghan says:
    05/05/2014 at 4:04 pm

    應該是orthonormal matrix?

    Reply
  2. ccjou says:
    05/06/2014 at 8:18 am

    確實是orthogonal matrix，它的列或行向量構成一orthonormal set，但不存在orthonormal matrix這個詞。

    Reply
  3. Regan says:
    05/07/2014 at 12:26 am

    There is no standard terminology for these matrices. They are sometimes called “orthonormal matrices”, sometimes “orthogonal matrices”, and sometimes simply “matrices with orthonormal rows/columns”.
    http://en.wikipedia.org/wiki/Orthogonal_matrix

    Reply
    • ccjou says:
      05/07/2014 at 9:01 am

      這是斷章取義，引文的these matrices是指非方陣。正交(orthogonal)矩陣是方陣，它是一個通用的名稱。

      Reply
      • Regan Yuan says:
        05/08/2014 at 12:43 pm

        老师您好，请问如果一个m*n的矩阵A（m不等于n），A的第i个row vector都与另一个矩阵B（n*m）的第i个column vector“正交”，dot product is 1.而其余的dot product is zero.这样A 与B应该称为什么矩阵呢？谢谢

        Reply

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