7.5 QR因数化
章节大纲
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When we discussed diagonalization we essentially taught ourselves how to factor matrices. In this lesson we are going to apply our knowledge of the Gram-Schmidt process and orthonormalization in order to factor matrices by another method, known as QR factorization.
::当我们讨论对等化时,我们基本上教自己如何计算矩阵。在这个教训中,我们将运用我们关于Gram-Schmidt过程和正正正化的知识,以便用另一种方法,即QR系数化来计算矩阵。
Theorem: Given an matrix with linearly independent columns one can factor that matrix into the form where is also an matrix with columns which form an orthonormal basis for the column space of and is an upper triangular matrix which is invertible and has positive entries on the diagonal.
::论理:如果有一个有线性独立的列的 mxn 矩阵A,那么可以将该矩阵纳入表AR,其中Q也是mxn 矩阵,其列构成A和R列空间的正正态基数,是 nxn 的上三角矩阵,不可逆,在对角有正条目。
Try to prove this on your own using properties about orthogonal matrices and the Gram-Schmidt process. I will just run through an example of this technique being used.
::尝试使用关于矩形矩阵的属性和 Gram-Schmidt 进程来证明这一点。 我将尝试一个使用这种技术的例子 。Example: Let
::示例:让我们A=[100110111111]Applying the Gram-Schmidt process to get an orthonormal basis for the column space of A, we get that
::应用 Gram- Schmidt 进程来获得 A 列空间的正正异常值, 我们知道
::[12-312012112-2612112161212161216]Now, we can solve for taking because Q is an orthogonal matrix.
::现在,我们可以解决 R 使用 QTAT( QR) =IR=R 的问题, 因为 Q 是正对矩阵 。Then, we get that
::然后,我们得到
::R=[121212121212-3121121212120-261616]][100111111]R=[232103122120026]
::因此,我们找到了我们的Q和R。
::欲了解更多信息,请查看这些来源,这些来源将为您提供关于QR的宝贵见解。