Regarding the elimination of zero complex terms from State Transition Matrix

I have a Matrix A defined as

A1 = [-(1/2)*(1+(1/sqrt(2))) 1/4;-(1/2) -(1/2)*(1-(1/sqrt(2)))];

which is equivalent to

A2 = [-0.8536 0.2500; -0.5000 -0.1464];

But when I take eigenvalues in both cases I get different eigenvalues

>> eig(A1)

ans =

-0.5000 + 0.0000i

-0.5000 - 0.0000i

eigenvaules are repeated, but MATLAB considering these as distinct roots(Complex conjugate)

>> eig(A2)

ans =

-0.5057

-0.4943

because of truncation, roots seems to be Different.

I have no problem with A2 matrix. But I want the system to consider only real part of eigenvalues of A1 matrix. Because of +0.0000i and -0.0000i the equations which depend on eigenvalues of A is changing.

I have already used real(egg(A1)) but I wanted to Calculate the state transition matrix i.e. e^(At)

syms t

phi = vpa(expm(A*t),4)

in this expression it should take those repeated roots of -0.5 and -0.5. But it is not taking.

So, please help me out.

Thank You.

Answer 1

[Mark Sherstan]

real(eig(A1))

[Rajani Metri]

Thank you, but I already used this command.

I wanted to calculate the e^(At) matrix.

For which I used following commands:

syms t

phi = vpa(expm(A*t),4)

but it still takes conj values.

[yitao yitao]

thanks

تشخیص سرقت ادبی انلاین

Answer 2

[madhan ravi]

The imaginary part is not zero:

"Ideally, the eigenvalue decomposition satisfies the relationship. Since eig performs the decomposition using floating-point computations, then A*V can, at best, approach V*D. In other words, A*V - V*D is close to, but not exactly, 0." mentioned here

>> A1  =  [-(1/2)*(1+(1/sqrt(2))) 1/4;-(1/2) -(1/2)*(1-(1/sqrt(2)))];
   vpa(eig(A1))
 
ans =
 
 - 0.5 + 0.0000000064523920698794617994748209544899i
 - 0.5 - 0.0000000064523920698794617994748209544899i
 
>> 

[Rajani Metri]

Yes, I know.

That's what I want to eliminate those very small complex terms from the e^At expression. I want those roots to be act as repeated roots -0.5, -0.5.

Thank you.

[madhan ravi]

Just give an example of your desired output

[Rajani Metri]

I calculated the e^At (state transition matrix) on paper and it comes out as follows:

[(e^(-0.5t)) - 0.3536*t*(e^(-0.5t)), 0.25*t*(e^(-0.5t));

-0.5*t*(e^(-0.5t)), (e^(-0.5t)) + 0.3536*t*(e^(-0.5t))]

it is a 2x2 matrix.