Matlab 建立符号正交矩阵原型的最简单方法
我有8个sins和cosines，我试图用Matlab符号化地定义它们，如下所示。我的目标是用所有这些sins和cosines符号化地构建一个8x8的矩阵H（累计Givens旋转矩阵），最后看看这个H正交投影矩阵的公式是什么。在概念上，我可以使用下面的代码Matlab 建立符号正交矩阵原型的最简单方法,matlab,math,matrix,linear-algebra,symbolic-math,Matlab,Math,Matrix,Linear Algebra,Symbolic Math,我有8个sins和cosines，我试图用Matlab符号化地定义它们，如下所示。我的目标是用所有这些sins和cosines符号化地构建一个8x8的矩阵H（累计Givens旋转矩阵），最后看看这个H正交投影矩阵的公式是什么。在概念上，我可以使用下面的代码G7*G6*…*G0*I，其中I是标识8x8，Gi是对应于元素的Givens旋转（I:I+1，I:I+1） 代码失败，出现以下错误，无法找到修复方法： The following error occurred converting from s
G7*G6*…*G0*IThe following error occurred converting from sym to double:
Error using mupadmex
Error in MuPAD command: DOUBLE cannot convert the input expression into a double array.
If the input expression contains a symbolic variable, use the VPA function instead.
Error in build_rotH_test (line 26)
H(1:2,1:2) = [c_0 -s_0; s_0 c_0]
我是这样解决的。注：我意识到我需要每个旋转的转置，这样我就可以构建和应用H'*x，即G7'*G6'*…*G0'*I，这就是为什么在解决方案中翻转正弦符号的原因
clear all;
% defining 0 and 1 as symbols too, solves the problem
sym_0 = sym('0');
sym_1 = sym('1');
c0 = sym('c0');
c1 = sym('c1');
c2 = sym('c2');
c3 = sym('c3');
c4 = sym('c4');
c5 = sym('c5');
c6 = sym('c6');
c7 = sym('c7');
s0 = sym('s0');
s1 = sym('s1');
s2 = sym('s2');
s3 = sym('s3');
s4 = sym('s4');
s5 = sym('s5');
s6 = sym('s6');
s7 = sym('s7');
% create H orthogonal matrix using the sin and cos symbols
% filling in the first rotation
I = repmat(sym_0,9,9);
for i=1:9
I(i,i)=sym_1;
end
H = I
H(1:2,1:2) = [c0 s0; -s0 c0]
% build the 2nd rotation and update H
G = I;
G(2:3,2:3) = [c1 s1; -s1 c1]
H = G*H;
% build the 3rd rotation and update H
G = I;
G(3:4,3:4) = [c2 s2; -s2 c2]
H = G*H;
% build the 4rth rotation and update H
G = I;
G(4:5,4:5) = [c3 s3; -s3 c3]
H = G*H;
% build the 5th rotation and update H
G = I;
G(5:6,5:6) = [c4 s4; -s4 c4]
H = G*H;
% build the 6th rotation and update H
G = I;
G(6:7,6:7) = [c5 s5; -s5 c5]
H = G*H;
% build the 7th rotation and update H
G = I;
G(7:8,7:8) = [c6 s6; -s6 c6]
H = G*H;
% build the 8th rotation and update H
G = I;
G(8:9,8:9) = [c7 s7; -s7 c7]
H = G*H
H =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
H =
[ c0, s0, 0, 0, 0, 0, 0, 0, 0]
[ -s0, c0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, c1, s1, 0, 0, 0, 0, 0, 0]
[ 0, -s1, c1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, c2, s2, 0, 0, 0, 0, 0]
[ 0, 0, -s2, c2, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, c3, s3, 0, 0, 0, 0]
[ 0, 0, 0, -s3, c3, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, c4, s4, 0, 0, 0]
[ 0, 0, 0, 0, -s4, c4, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, c5, s5, 0, 0]
[ 0, 0, 0, 0, 0, -s5, c5, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, c6, s6, 0]
[ 0, 0, 0, 0, 0, 0, -s6, c6, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, c7, s7]
[ 0, 0, 0, 0, 0, 0, 0, -s7, c7]
H =
[ c0, s0, 0, 0, 0, 0, 0, 0, 0]
[ -c1*s0, c0*c1, s1, 0, 0, 0, 0, 0, 0]
[ c2*s0*s1, -c0*c2*s1, c1*c2, s2, 0, 0, 0, 0, 0]
[ -c3*s0*s1*s2, c0*c3*s1*s2, -c1*c3*s2, c2*c3, s3, 0, 0, 0, 0]
[ c4*s0*s1*s2*s3, -c0*c4*s1*s2*s3, c1*c4*s2*s3, -c2*c4*s3, c3*c4, s4, 0, 0, 0]
[ -c5*s0*s1*s2*s3*s4, c0*c5*s1*s2*s3*s4, -c1*c5*s2*s3*s4, c2*c5*s3*s4, -c3*c5*s4, c4*c5, s5, 0, 0]
[ c6*s0*s1*s2*s3*s4*s5, -c0*c6*s1*s2*s3*s4*s5, c1*c6*s2*s3*s4*s5, -c2*c6*s3*s4*s5, c3*c6*s4*s5, -c4*c6*s5, c5*c6, s6, 0]
[ -c7*s0*s1*s2*s3*s4*s5*s6, c0*c7*s1*s2*s3*s4*s5*s6, -c1*c7*s2*s3*s4*s5*s6, c2*c7*s3*s4*s5*s6, -c3*c7*s4*s5*s6, c4*c7*s5*s6, -c5*c7*s6, c6*c7, s7]
[ s0*s1*s2*s3*s4*s5*s6*s7, -c0*s1*s2*s3*s4*s5*s6*s7, c1*s2*s3*s4*s5*s6*s7, -c2*s3*s4*s5*s6*s7, c3*s4*s5*s6*s7, -c4*s5*s6*s7, c5*s6*s7, -c6*s7, c7]
我是这样解决的。注：我意识到我需要每个旋转的转置，这样我就可以构建和应用H'*x，即G7'*G6'*…*G0'*I，这就是为什么在解决方案中翻转正弦符号的原因
clear all;
% defining 0 and 1 as symbols too, solves the problem
sym_0 = sym('0');
sym_1 = sym('1');
c0 = sym('c0');
c1 = sym('c1');
c2 = sym('c2');
c3 = sym('c3');
c4 = sym('c4');
c5 = sym('c5');
c6 = sym('c6');
c7 = sym('c7');
s0 = sym('s0');
s1 = sym('s1');
s2 = sym('s2');
s3 = sym('s3');
s4 = sym('s4');
s5 = sym('s5');
s6 = sym('s6');
s7 = sym('s7');
% create H orthogonal matrix using the sin and cos symbols
% filling in the first rotation
I = repmat(sym_0,9,9);
for i=1:9
I(i,i)=sym_1;
end
H = I
H(1:2,1:2) = [c0 s0; -s0 c0]
% build the 2nd rotation and update H
G = I;
G(2:3,2:3) = [c1 s1; -s1 c1]
H = G*H;
% build the 3rd rotation and update H
G = I;
G(3:4,3:4) = [c2 s2; -s2 c2]
H = G*H;
% build the 4rth rotation and update H
G = I;
G(4:5,4:5) = [c3 s3; -s3 c3]
H = G*H;
% build the 5th rotation and update H
G = I;
G(5:6,5:6) = [c4 s4; -s4 c4]
H = G*H;
% build the 6th rotation and update H
G = I;
G(6:7,6:7) = [c5 s5; -s5 c5]
H = G*H;
% build the 7th rotation and update H
G = I;
G(7:8,7:8) = [c6 s6; -s6 c6]
H = G*H;
% build the 8th rotation and update H
G = I;
G(8:9,8:9) = [c7 s7; -s7 c7]
H = G*H
H =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
H =
[ c0, s0, 0, 0, 0, 0, 0, 0, 0]
[ -s0, c0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, c1, s1, 0, 0, 0, 0, 0, 0]
[ 0, -s1, c1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, c2, s2, 0, 0, 0, 0, 0]
[ 0, 0, -s2, c2, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, c3, s3, 0, 0, 0, 0]
[ 0, 0, 0, -s3, c3, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, c4, s4, 0, 0, 0]
[ 0, 0, 0, 0, -s4, c4, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, c5, s5, 0, 0]
[ 0, 0, 0, 0, 0, -s5, c5, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, c6, s6, 0]
[ 0, 0, 0, 0, 0, 0, -s6, c6, 0]
[ 0, 0, 0, 0, 0, 0, 0, 0, 1]
G =
[ 1, 0, 0, 0, 0, 0, 0, 0, 0]
[ 0, 1, 0, 0, 0, 0, 0, 0, 0]
[ 0, 0, 1, 0, 0, 0, 0, 0, 0]
[ 0, 0, 0, 1, 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0, 1, 0, 0, 0]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0]
[ 0, 0, 0, 0, 0, 0, 0, c7, s7]
[ 0, 0, 0, 0, 0, 0, 0, -s7, c7]
H =
[ c0, s0, 0, 0, 0, 0, 0, 0, 0]
[ -c1*s0, c0*c1, s1, 0, 0, 0, 0, 0, 0]
[ c2*s0*s1, -c0*c2*s1, c1*c2, s2, 0, 0, 0, 0, 0]
[ -c3*s0*s1*s2, c0*c3*s1*s2, -c1*c3*s2, c2*c3, s3, 0, 0, 0, 0]
[ c4*s0*s1*s2*s3, -c0*c4*s1*s2*s3, c1*c4*s2*s3, -c2*c4*s3, c3*c4, s4, 0, 0, 0]
[ -c5*s0*s1*s2*s3*s4, c0*c5*s1*s2*s3*s4, -c1*c5*s2*s3*s4, c2*c5*s3*s4, -c3*c5*s4, c4*c5, s5, 0, 0]
[ c6*s0*s1*s2*s3*s4*s5, -c0*c6*s1*s2*s3*s4*s5, c1*c6*s2*s3*s4*s5, -c2*c6*s3*s4*s5, c3*c6*s4*s5, -c4*c6*s5, c5*c6, s6, 0]
[ -c7*s0*s1*s2*s3*s4*s5*s6, c0*c7*s1*s2*s3*s4*s5*s6, -c1*c7*s2*s3*s4*s5*s6, c2*c7*s3*s4*s5*s6, -c3*c7*s4*s5*s6, c4*c7*s5*s6, -c5*c7*s6, c6*c7, s7]
[ s0*s1*s2*s3*s4*s5*s6*s7, -c0*s1*s2*s3*s4*s5*s6*s7, c1*s2*s3*s4*s5*s6*s7, -c2*s3*s4*s5*s6*s7, c3*s4*s5*s6*s7, -c4*s5*s6*s7, c5*s6*s7, -c6*s7, c7]
我最初的想法是，你得到的错误是因为H是一个双精度的，试着做H=sym（I）；我不知道这是否能使一切正常，但它应该能消除您的错误。谢谢。我已经弄明白了。我会尽快发布解决方案，只要我被允许。问题是Matlab不允许将符号变量与实际值（由eye生成）组合在一起，所以我也将0和1构建为变量，然后它就可以正常工作了；我不知道这是否能使一切正常，但它应该能消除您的错误。谢谢。我已经弄明白了。我会尽快发布解决方案，只要我被允许。问题是Matlab不允许将符号变量与实际值（由eye生成）组合在一起，所以我也将0和1构建为变量，然后它就可以正常工作了。