clear
format compact

%%
% Define the vectors 
m = 4;
z = ones(m, 1);  % ||z|| = 2
e1 = eye(m, 1);

%%
% Applying the Householder definitions gives us
v = norm(z)*e1 - z;
P = eye(m) - 2/(v'*v)*(v*v');   % reflector

%%
% By design we can use the reflector to get the structure we seek:
P*z

%%
% Verify
norm(P*z - e1*norm(z))