理顺 QR 分解算法

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1. Why QR

When A and b are known, to solver the minimization of , where .

The reduction of A to various canonical form via orthogonal transformations should use Householder reflections and Givens rotations.

2. preview on orthogonal matrix

2.1 Orthogonal matrix

is  orthogonal matrix, if:

2.2 rotation matrix is orthogonal matrix

If

is orthogonal and   is a rotation matrix.

If , then is obtained by rotating counterclockwise through an angle .

2.3 reflection matrix is orthogonal matrix

is orthogonal and is a reflection matrix.

If , then is obtained by reflecting the vector across the line defined by

That means and are axial symmetry by S,

x is the preimage, y is the image, S is the mirror surface.

looks like :

%input x, ta = theta

%x = [-sqrt(2)/2.0, sqrt(2)/2.0]
x = [1; 1;]
ta = pi/5

S = [cos(ta/2.0), sin(ta/2.0)]
Q = [cos(ta), sin(ta); sin(ta), -cos(ta);]

y = Q*x

figure;
%1. draw axis
xmin = -2
xmax = 2
ymin = -2
ymax = 2

axisx = xmin:xmax;
axisy = zeros(size(axisx));
plot(axisx, axisy, 'k--', 'LineWidth', 0.7); % Plot x=0 axis
hold on;
plot(axisy, xmin:xmax, 'k--', 'LineWidth', 0.7); % Plot y=0 axis
hold on;

%2. draw surface of mirror
sx = -2*S(1):0.5:2*S(1)
sy = (S(2)/S(1))*sx
plot(sx, sy)
text(sx,sy, 'S')
hold on;

%3. draw preimage
plot(x(1), x(2), 'ro')

text(x(1)+0.1, x(2)+0.1, 'x')
hold on;

%4. draw image
plot(y(1), y(2), 'bo')
text(y(1)+0.1, y(2)+0.1, 'y')

%5. axis label
xlabel("X")
ylabel('Y')
v=[xmin, xmax, ymin, ymax]
axis(v)
%axis on

3. Householder transformation

In section 2, the reflection is introduced from the mirror surface. But, in this section, it is introduced from normal direction.

Let

then is

a    

or   

or 

which are synonyms.

And is the .

When

is the image from by reflecting with the hyuperplane and the mirror surface is cross the point.

If

let   (this is the normal direction)

then ;

约化定理：

Let ,

then st.

and:

约化定理毕；

约化定理example：

Let then , and

to calculate :

then 

Here is the matlab code:

reduce_01.m:

x=[3;5;1;1;]
sigmaa =sign(x(1))*norm(x)
u = x+sigmaa*eye(4)(:,1)
betaa = 0.5*(norm(u))^2
H = eye(4) - (1.0/betaa)*u*u'
%debug
%h=betaa*H

y = H*x

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