每週問題 May 5, 2014

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^TCCC^T 有相同的特徵值,推論 I_n-C^TCI_n-CC^T 也有相同的特徵值,也就是說,A^TADD^T 有相同的特徵值,故 AD 有相同的奇異值。

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8 Responses to 每週問題 May 5, 2014

  1. Shenghan says:

    應該是orthonormal matrix?

  2. ccjou says:

    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。

  3. Regan says:

    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

    • ccjou says:

      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。

      • Regan Yuan says:

        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

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