# $QR$ Factorization

## Reduced $QR$ Factorization

Consider a _flag_ of subspaces generated by linearly independent vectors: 

$$\langle \mathbf{a}_1\rangle \subset \langle \mathbf{a}_1,\mathbf{a}_2\rangle \subset \cdots$$

The idea of a $QR$-factorization is the construction of a sequence of _orthonormal_ vectors $\mathbf{q}_1$, $\mathbf{q}_2$, $\mathbf{q}_3$,... that span the above successive subspaces.

That is, for each integer $j$, we have:

$$
\langle \mathbf{a}_1,\dots,\mathbf{a}_j\rangle = \langle\mathbf{q}_1,\dots, \mathbf{q}_j\rangle.
$$

In other words:

$$
A= \left[\mathbf{a}_1 \mid \cdots \mid \mathbf{a}_n\right] = \left[\mathbf{q}_1\mid \cdots \mid \mathbf{q}_n\right]\cdot \begin{bmatrix}r_{11} & r_{12} & \cdots & r_{1n}\\ & r_{22} &  & \vdots \\ & & \ddots & \vdots \\ & & & r_{nm}\end{bmatrix} = Q\cdot R
$$

Writing this out as a system of linear equations:
\begin{align}\mathbf{a}_1 &= r_{11} \mathbf{q}_1,\\ \mathbf{a}_2 &= r_{12}\mathbf{q}_1 + r_{22} \mathbf{q}_2,\\ \mathbf{a}_3 &= r_{13} \mathbf{q}_1 + r_{23} \mathbf{q}_2 + r_{33}\mathbf{q}_3,\\ & \vdots \\ \mathbf{a}_n &= r_{1n}\mathbf{q}_1 + \cdots + r_{nn} \mathbf{q}_n.\end{align}

In matrix form: 

$$ A = \hat{Q}\cdot \hat{R}$$ 

where $\hat{Q}$ is an $m\times n$ matrix with orthonormal columns, and $\hat{R}$ is an $n\times n$ upper-triangular matrix.  This is the so-called _reduced $QR$-factorization_.


```julia
A = [1 2 3; 4 5 6]
Q,R = qr(A)
qrfact(A), Q, R
```




    (Base.LinAlg.QRCompactWY{Float64,Array{Float64,2}}([-4.12311 -5.33578 -6.54846; 0.780776 -0.727607 -1.45521],[1.24254 0.0; 6.92462e-310 0.0]),
    [-0.242536 -0.970143; -0.970143 0.242536],
    
    [-4.12311 -5.33578 -6.54846; 0.0 -0.727607 -1.45521])



## Full $QR$ Factorization

A _full $QR$ factorization_ of $A\in\mathbb{C}^{m\times n}$ ($m\geq n$) appends the $m-n$ orthonormal colums to $\hat{Q}$ making it $m\times m$.

Notice, in the full $QR$ factorization, the columns $\mathbf{q}_j$ for $j>n$, are orthogonal to $C(A)$.  Assuming $A$ is of full rank $n$, they form a basis for $N(A^*)$.

** Nota Bene: ** The $QR$ factorization is to matrices what polar coordinates is to vectors: 

$$Q\cdot R = \text{[orthogonal matrix]}\cdot \text{[positive definite matrix]}$$

## Gram-Schmidt Orthonormalization

Given $\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3,\dots$, we can construct $\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3,\dots$ via a process of successive orthogonalization.

To compute the $j$-th orthogonal vector we compute:

$$\mathbf{v}_j = \mathbf{a}_j - (\mathbf{q}_1^*\mathbf{a}_j)\mathbf{q}_1 -(\mathbf{q}_2^*\mathbf{a}_j)\mathbf{q}_2 - \cdots - (\mathbf{q}_{j-1}^*\mathbf{a}_j) \mathbf{q}_{j-1}.$$

Then set: $$\mathbf{q}_j := \frac{\mathbf{v}_j}{\|\mathbf{v}_j\|}.$$




```julia
m = 5;
n = 3;
A = rand(m,n);
Q = zeros(m,n);  R = zeros(n,n); 
R[1,1] = norm(A[:,1]);
Q[:,1] = A[:,1]/R[1,1];
@time for j = 2:n
    R[1:j-1,j] = Q[:,1:j-1]'*A[:,j];
    v = A[:,j] - Q[:,1:j-1]*R[1:j-1,j];
    R[j,j] = norm(v);
    Q[:,j] = v/R[j,j];
end
A, Q, R, Q*R
```

      0.000042 seconds (49 allocations: 2.906 KB)





    (
    [0.796102 0.135397 0.0502899; 0.666858 0.143114 0.721173; … ; 0.350599 0.75058 0.841905; 0.052776 0.644623 0.256538],
    
    [0.722889 -0.20077 -0.625705; 0.605531 -0.143994 0.558806; … ; 0.318357 0.474183 0.44998; 0.0479225 0.50426 -0.192359],
    
    [1.10128 0.528395 0.784844; 0.0 1.22814 0.667665; 0.0 0.0 0.612137],
    
    [0.796102 0.135397 0.0502899; 0.666858 0.143114 0.721173; … ; 0.350599 0.75058 0.841905; 0.052776 0.644623 0.256538])



## Existence and uniqueness

> ** THEOREM. [Existence] ** Every $A\in \mathbb{C}^{m\times n}$, $m\geq n$, has a full $QR$ factorization.

** Proof. ** If $A$ has full rank, then the Gram-Schmidt process produces a reduced $QR$ factorization — since the only way the Gram-Schmidt fails is if $\mathbf{v}_j=\mathbf{0}$, however this implies $$\mathbf{a}_j \in \langle \mathbf{q}_1,\dots, \mathbf{q}_{j-1}\rangle=\langle \mathbf{a}_1,\dots, \mathbf{a}_{j-1}\rangle.$$

If $A$ is _not_ of full rank, then $\mathbf{v}_j=0$ for some $j$, simply pick any unit $\mathbf{q}_j$ in the orthogonal complement of $\langle \mathbf{q}_1,\dots, \mathbf{q}_{j-1}\rangle$.

For the full $QR$ factorization simply add arbitrary orthonormal vectors in the orthogonal complement.

> ** THEOREM. [Uniqueness] ** Each $A\in\mathbb{C}^{m\times n}$, $m\geq n$, of full rank has a unique reduced $QR$ factorization $A = \hat{Q} \hat{R}$ with $r_{jj}>0$.

** Proof. ** From the equations defining the Gram-Schmidt and the assumption of full rank (i.e. $\mathbf{v}_j\ne 0$), we see that the only ambiguity is the sign of $r_{jj}$.

## Continuous functions as vectors

Suppose $\mathbf{a}_j(x) \in L^2[-1,1]$ are now continuous (square-integrable) functions:
$$
A = \left[ 1 \mid x \mid x^2 \mid \cdots \mid x^{n-1}\right].
$$

$$
A = QR =\left[ \mathbf{q}_0(x) \mid \mathbf{q}_1(x) \mid \cdots \mid \mathbf{q}_{n-1}(x)\right] \begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n}\\ & r_{22} & & \vdots \\ & & \ddots & \vdots\\ & & & r_{nn}\end{bmatrix}
$$

Where the columns $\mathbf{q}_j(x)$ are orthonormal with respect to the inner product:

$$
\int_{-1}^1 \overline{\mathbf{q}_i(x)}\mathbf{q}_j(x) dx = \delta_{ij} = \begin{cases} 1 & \text{if } i=j,\\ 0 & \text{if } i\ne j.\end{cases}
$$

From the Gram-Schmidt process, we know that $\mathbf{q}_j$ is a polynomial of degree $j$.

The will be scalar multiples of the _Legendre polynomials_ $P_j$:
\begin{align}
P_0(x) &= 1\\ 
P_1(x) &= x\\
P_2(x) &= \frac{3}{2}x^2-\frac{1}{2}\\
P_3(x) &= \frac{5}{2}x^3-\frac{3}{2}x.
\end{align}

These polynomials are orthonormal, and normalized so that $P_j(1)=1$.


```julia
using Gadfly

f_0(x) = 1;
f_1(x) = x;
f_2(x) = (3/2)x^2-(1/2);
f_3(x) = (5/2)x^3-(3/2)x;

plot([f_0, f_1, f_2, f_3],-1,1)
```




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> _**Question.**  What is the "projection matrix" $\hat{Q}\hat{Q}^*$ associated with $\hat{Q}$?_

$$
L^2[-1,1] \to L^2[-1,1]: f(\cdot) \mapsto \sum_{j=0}^{n-1} q_j(\cdot) \int_{-1}^1 \overline{q_j(x)} f(x) dx.
$$

The collection $\{P_j(x)\}$ provides a _spectral_ basis for the space of polynomials.

It turns out that the space $L_{\text{periodic}}^2[-\pi,\pi]$ has an orthonormal basis consisting of $\{e^{2\pi i n x}\}$.

This fact is the basis of _Fourier Theory_.   In particular, it says that if $f(x)=f(x+2\pi)$ is in $L^2[-\pi,\pi]$, then: 

$$f(x) = \sum_{n=0} c_n e^{2\pi in z}.$$

Where the Fourier coefficients can be computed as:
$$
c_n = \int_{-\pi}^\pi f(x) e^{-2\pi n ix} dx.
$$

## Solving $A\mathbf{x} = \mathbf{b}$ via QR factorization

Given a nonsingular $A\in \mathbb{C}^{m\times m}$, with $QR$ factorization $A=QR$.  We propose the following algorithm for solving: 

$$A \mathbf{x} = \mathbf{b}.$$

1. Compute the QR factorization $A=QR$.
- Compute $\mathbf{y}=Q^* \mathbf{b}$.
- Solve $R\mathbf{x} = \mathbf{y}$ for $\mathbf{x}$.

021a85736602f9ac2b07870b627262012e125478
ipynb
Jupyter Notebook
QR Factorization.ipynb
Twelve33/NumericalLinearAlgebra
4122cf464712855f81be82eb0e92de27aad1ea31
MIT
2020-05-23T23:55:16.000Z
2020-05-23T23:55:16.000Z
QR Factorization.ipynb
Twelve33/NumericalLinearAlgebra
4122cf464712855f81be82eb0e92de27aad1ea31
MIT
2019-04-06T04:19:20.000Z
2019-04-28T05:56:52.000Z
QR Factorization.ipynb
Twelve33/NumericalLinearAlgebra
4122cf464712855f81be82eb0e92de27aad1ea31
MIT
2019-04-06T04:04:45.000Z
2019-04-06T04:04:45.000Z
Qwen/Qwen-72B
1. YES
2. YES