## 每週問題 May 5, 2014

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.

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

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

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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

• ccjou says:

如果AB=I，則A是B的左逆，B是A的右逆。

• Regan Yuan says:

如此说来，逆矩阵是在A,B都是方阵前提下的一种特殊情况了。而左、右逆矩阵，都是逆矩阵的推广了？谢谢

• ccjou says:

是這樣。