数学代写|数值分析代写numerical analysis代考|MATHS7104

数学代写|数值分析代写numerical analysis代考|The Moore–Penrose Pseudo-inverse

If $\mathrm{A} \in \mathbb{C}^{m \times n}$ with $m \geq n=\operatorname{rank}(\mathrm{A})$, then there is a unique least squares solution to the system $A x=f$, which can be found by solving the normal equations or by computing the reduced $Q R$ factorization of $A$.

The question we want to address now is: What happens if $m \geq n>r=\operatorname{rank}(A)$ ? In this case, we say that $\mathrm{A}$ is rank deficient and we know that $\operatorname{ker}(\mathrm{A}) \neq{0}$. We have shown the existence of a least squares solution, even in this case. However, the solution will not be unique. Indeed, if $\boldsymbol{x}$ is a least squares solution and $\boldsymbol{\xi} \in \operatorname{ker}(\mathrm{A})$ is nonzero, then
$$\mathrm{A}^{\mathrm{H}} \boldsymbol{f}=\mathrm{A}^{\mathrm{H}} \mathrm{A} \boldsymbol{x}=\mathrm{A}^{\mathrm{H}} \mathrm{A}(\boldsymbol{x}+\boldsymbol{\xi}),$$
so that $x+\boldsymbol{\xi}$ is also a least squares solution. In other words, for every $\boldsymbol{\xi} \in \operatorname{ker}(\mathrm{A})$ we have
$$\Phi(\boldsymbol{x}+\boldsymbol{\xi})=|\boldsymbol{f}-\mathrm{A}(\boldsymbol{x}+\boldsymbol{\xi})|_2^2=|\boldsymbol{f}-\mathrm{A} \boldsymbol{x}|_2^2=\Phi(\boldsymbol{x}) .$$
To be able to remove the nonuniqueness, we will require the solution to be, in a sense, the smallest.

Definition $5.37$ (minimum norm least squares solution). Let $\mathrm{A} \in \mathbb{C}^{m \times n}$ with $m \geq$ $n>r=\operatorname{rank}(\mathrm{A})$ and $f \in \mathbb{C}^m$. Define
$$\Phi(x)=|\boldsymbol{f}-\mathrm{Ax}|_2^2 .$$
The minimum norm least squares solution of $A x=f$ is $\hat{x} \in \mathbb{C}^n$ that satisfies:

1. $\hat{x}$ is a least squares solution, i.e., $\Phi(\hat{x}) \leq \Phi(x)$ for all $x \in \mathbb{C}^n$.
2. If $\Phi(\hat{x})=\Phi(\boldsymbol{x})$, then $|\hat{x}|_2 \leq|\boldsymbol{x}|_2$.
We find this minimum norm solution using the so-called pseudo-inverse of $A$, which was introduced in Problem 2.5. We recall here its definition and basic properties for convenience.

数学代写|数值分析代写numerical analysis代考|The Modified Gram–Schmidt Process

Let us redefine the Gram-Schmidt process using the language of orthogonal projection matrices. To do so, for $k=1, \ldots, n$, we define the matrix $P_k \in \mathbb{C}^{n \times n}$ by its action on a vector $\boldsymbol{w} \in \mathbb{C}^n$,
$$\mathrm{P}k \boldsymbol{w}=\boldsymbol{w}-\sum{j=1}^{k-1}\left(\boldsymbol{w}, \boldsymbol{q}j\right)_2 \boldsymbol{q}_j .$$ where $\left{\boldsymbol{q}_1, \ldots, \boldsymbol{q}{k-1}\right}$ is an orthonormal set.

Definition $5.41$ (modified Gram-Schmidt). Suppose that $S=\left{a_1, \ldots, a_k\right} \subset \mathbb{C}*^n$ with $k \leq n$. The modified Gram-Schmidt process is an algorithm for generating the set of vectors $Q=\left{\boldsymbol{q}_1, \ldots, \boldsymbol{q}_k\right}$ recursively as follows: for $m=1$, $$q_1=\frac{1}{\left|a_1\right|_2} a_1 .$$ For $2 \leq m \leq k$, suppose that $\left{\boldsymbol{q}_1, \ldots, \boldsymbol{q}{m-1}\right}$ have been computed. Set
\begin{aligned} & \boldsymbol{v}m^1=\boldsymbol{a}_m, \ & \boldsymbol{v}_m^2=\mathrm{P}{\boldsymbol{q}1^{\perp}} \boldsymbol{v}_m^1, \ & \vdots \ & \boldsymbol{v}_m=\boldsymbol{v}_m^m=\mathrm{P}{\boldsymbol{q}_{m-1}^{\perp}} \boldsymbol{v}_m^{m-1} . \end{aligned}
If $\boldsymbol{v}_m=\mathbf{0}$, the process terminates. Otherwise, the process continues with
$$\boldsymbol{q}_m=\frac{1}{\left|\boldsymbol{v}_m\right|_2} \boldsymbol{v}_m .$$
As noted in Remark $5.31$ the classical Gram-Schmidt algorithm is unstable. But it turns out that the modified Gram-Schmidt process is stable and is the one that is used in practical computations. Since it results from only cosmetic changes to the definition of the original Gram-Schmidt process, we have the following result.

数值分析代考

数学代写|数值分析代写numerical analysis代考|The Moore–Penrose Pseudo-inverse

$$\mathrm{A}^{\mathrm{H}} \boldsymbol{f}=\mathrm{A}^{\mathrm{H}} \mathrm{A} \boldsymbol{x}=\mathrm{A}^{\mathrm{H}} \mathrm{A}(\boldsymbol{x}+\boldsymbol{\xi}),$$

$$\Phi(\boldsymbol{x}+\boldsymbol{\xi})=|\boldsymbol{f}-\mathrm{A}(\boldsymbol{x}+\boldsymbol{\xi})|_2^2=|\boldsymbol{f}-\mathrm{A} \boldsymbol{x}|_2^2=\Phi(\boldsymbol{x}) .$$

$$\Phi(x)=|\boldsymbol{f}-\mathrm{Ax}|_2^2 .$$

1. $\hat{x}$ 是最小二乘解，即 $\Phi(\hat{x}) \leq \Phi(x)$ 对所有人 $x \in \mathbb{C}^n$.
2. 如果 $\Phi(\hat{x})=\Phi(\boldsymbol{x})$ ，然后 $|\hat{x}|_2 \leq|\boldsymbol{x}|_2$.
我们使用所谓的伪逆来找到这个最小范数解 $A$ ，这是在问题 $2.5$ 中引入的。为了方便起见，我 们在这里回顾它的定义和基本属性。

数学代写|数值分析代写numerical analysis代考|The Modified Gram–Schmidt Process

$$\mathrm{Pk} \boldsymbol{w}=\boldsymbol{w}-\sum j=1^{k-1}(\boldsymbol{w}, \boldsymbol{q} j)2 \boldsymbol{q}_j .$$ 在哪里 Yeft{ $\backslash$ boldsymbol{q}_1, \Idots, \boldsymbol{q}{k-1}\right } } \text { 是正交集。 } 定义5.41 (修改后的 Gram-Schmidt)。假设 $S=\backslash$ left{a_1, \Idots, a_k\right } } \text { subset } \backslash m \text { mathbb } { C } ^ { \star \wedge } n \text { 和 } $k \leq n$. 改进的 Gram-Schmidt 过程是一种用于生成向量集的算法 $$q_1=\frac{1}{\left|a_1\right|_2} a_1$$ $$\boldsymbol{v} m^1=\boldsymbol{a}_m, \quad \boldsymbol{v}_m^2=\mathrm{P} \boldsymbol{q} \mathbf{1}^{\perp} \boldsymbol{v}_m^1, \vdots \quad \boldsymbol{v}_m=\boldsymbol{v}_m^m=\mathrm{P} \boldsymbol{q}{m-1}^{\perp} \boldsymbol{v}_m^{m-1}$$

$$\boldsymbol{q}_m=\frac{1}{\left|\boldsymbol{v}_m\right|_2} \boldsymbol{v}_m$$

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