easiest way to prototype a symbolic orthogonal matrix - matlab

Question

I have 8 sins and cosines that I try to symbolically define as shown below using Matlab. My goal is to symbolically build a matrix H (accumulated Givens rotations matrix) of 8x8 using all these sins and cosines and end up seeing what the formula for this H orthogonal projection matrix is. I can do that using the code below conceptually G7*G6*...*G0*I where I is the Identity 8x8 and the Gi are the Givens rotation corresponding to elements (i:i+1,i:i+1).
c_0 = sym('c_0');
c_1 = sym('c_1');
c_2 = sym('c_2');
c_3 = sym('c_3');
c_4 = sym('c_4');
c_5 = sym('c_5');
c_6 = sym('c_6');
c_7 = sym('c_7');
s_0 = sym('s_0');
s_1 = sym('s_1');
s_2 = sym('s_2');
s_3 = sym('s_3');
s_4 = sym('s_4');
s_5 = sym('s_5');
s_6 = sym('s_6');
s_7 = sym('s_7');
% create H orthogonal matrix using the sin and cos symbols
% filling in the first rotation
I=eye(9,9)
H = I;
H(1:2,1:2) = [c_0 -s_0; s_0 c_0]
% build the 2nd rotation and update H
G = I;
G(2:3,2:3) = [c_1 -s_1; s_1 c_1]
H = G*H
% build the 3rd rotation and update H
G = I;
G(3:4,3:4) = [c_2 -s_2; s_2 c_2]
H = G*H
% build the 4rth rotation and update H
G = I;
G(4:5,4:5) = [c_3 -s_3; s_3 c_3]
H = G*H
% build the 5th rotation and update H
G = I;
G(5:6,5:6) = [c_4 -s_4; s_4 c_4]
H = G*H
% build the 6th rotation and update H
G = I;
G(6:7,6:7) = [c_5 -s_5; s_5 c_5]
H = G*H
% build the 7th rotation and update H
G = I;
G(7:8,7:8) = [c_6 -s_6; s_6 c_6]
H = G*H
% build the 8th rotation and update H
G = I;
G(8:9,8:9) = [c_7 -s_7; s_7 c_7]
H = G*H
The code fails with the following error and can't find how to fix this:
The 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]

Answer 1

I solved it like this. Note I realized I need the transpose of each rotation so I can build and apply H'*x i.e. G7'*G6'*...*G0'*I that's why the sin signs are flipped in the solution.
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
and the output is:
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]