## 每週問題 June 19, 2017

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.

$n\times n$ 階可逆矩陣 $A$ 的線性獨立行向量 (column vector) 為 $\mathbf{a}_1,\ldots,\mathbf{a}_n$，即 $A=\begin{bmatrix}\mathbf{a}_1&\cdots&\mathbf{a}_n\end{bmatrix}$。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

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

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