每週問題 June 19, 2017

證明一個可逆矩陣存在 QR 分解。

Prove that an invertible matrix A can be represented in the form A=QR, where Q is an orthogonal matrix and R is an upper triangular matrix.

 
參考解答:

n\times n 階可逆矩陣 A 的線性獨立行向量 (column vector) 為 \mathbf{a}_1,\ldots,\mathbf{a}_n。Gram-Schmidt 正交化定理表明存在一個單範正交 (orthonormal) 向量集合 \{\mathbf{q}_1,\ldots,\mathbf{q}_n\} 使得 \mathbf{q}_i\in\hbox{span}\{\mathbf{a}_1,\ldots,\mathbf{a}_i\}i=1,\ldots,n。換句話說,存在一個上三角矩陣 T 使得

Q=\begin{bmatrix} \mathbf{q}_1&\cdots&\mathbf{q}_n \end{bmatrix}=\begin{bmatrix} \mathbf{a}_1&\cdots&\mathbf{a}_n \end{bmatrix}T=AT

其中 Q^T=Q^{-1}。因為 QA 是可逆的,推論 T 是可逆的。因此,A=QR,其中 R=T^{-1} 是上三角矩陣。

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