How can I exchange sqrt(-1) with 1i

Hello all,

I want Matlab to show 1i instead of sqrt(-1). (Example see below)

I tried simplify, subs and other commands, however I cant get Matlab to exclude sqrt(-1) and show 1i.

syms g l K u
A_3 = [0 0 0; 0 0 1; 0 -g/l 0]
b_3 = [K; -K/l; 0]

% Ruhelagenvektor
x_0 = - A_3' * b_3 * u

% Eigenwerte des Systems
syms lambda
f = det(A_3 - lambda*eye(3)) == 0
lambda_s = solve(f,lambda)

% Edit: I am interested in lambda_s, sorry for the confusion!
% Eigenkreisfrequenz des Systems
% omega = simplify(lambda_s + [0 1i -1i]')

I hope you guys can help me with this.

Best regards

[Matt J]

Are you saying that you don't see 1i when you run the code? I do.

omega =
 
                         0
  (-g)^(1/2)/l^0.5000 - 1i
- (-g)^(1/2)/l^0.5000 + 1i

[Rik]

Which release are you using?

syms g l K u
A_3 = [0 0 0; 0 0 1; 0 -g/l 0];
b_3 = [K; -K/l; 0];

% Ruhelagenvektor
x_0 = - A_3' * b_3 * u;

% Eigenwerte des Systems
syms lambda
f = det(A_3 - lambda*eye(3)) == 0;
lambda_s = solve(f,lambda);

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

% Eigenkreisfrequenz des Systems
omega = simplify(lambda_s + [0 1i -1i]')

omega = 

[Felix Richter]

R2020b

To be clear, I want lambda_s to show sqrt(g/l)*1i instead of sqrt(-g/l).

Answer 1

[Paul]

I thought that all that would be needed is to assume that g is positive:

syms g positive
sqrt(-g)

ans = 

Alas, this approach doesn't help in this problem:

syms g l K u
assume(g,'positive')
A_3 = [0 0 0; 0 0 1; 0 -g/l 0];
b_3 = [K; -K/l; 0];
% Ruhelagenvektor
x_0 = - A_3' * b_3 * u;
% Eigenwerte des Systems
syms lambda
f = det(A_3 - lambda*eye(3)) == 0;
lambda_s = solve(f,lambda)

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

lambda_s = 

simplify(lambda_s)

ans = 

The solution lambda_s is actually given in terms of (-g)^(1/2). Interestingly

(-g)^(1/2)

ans = 

simplify((-g)^(1/2))

ans = 

I guess the sqrt() function is unambigous but ^(1/2) is not. Try a substitution

syms temp
subs(subs(lambda_s,(-g)^(1/2),temp),temp,sqrt(-g))

ans = 

Correction and addition

[Felix Richter]

So a double subs does the trick.

Thank your very much!

[Paul]

I tried a single subs first

subs(lambda_s,(-g)^(1/2),sqrt(-g))

but that didn't yield the desired result. I don't know why.

[Paul]

Correction. Looks like a single subs() will work. Not sure why I thought otherwise. Looks like expand() will work as well.

syms g l K u
assume(g,'positive')
A_3 = [0 0 0; 0 0 1; 0 -g/l 0];
b_3 = [K; -K/l; 0];
% Ruhelagenvektor
x_0 = - A_3' * b_3 * u;
% Eigenwerte des Systems
syms lambda
f = det(A_3 - lambda*eye(3)) == 0;
lambda_s = solve(f,lambda)

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

lambda_s = 

subs(lambda_s,(-g)^(1/2),sqrt(-g))

ans = 

expand(lambda_s)

ans =