# How to find State Transition Matrix in terms of sin and cos if eigenvalues are Complex Conjugate?

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Rajani Metri on 23 Mar 2021
Commented: Star Strider on 24 Mar 2021
Hello,
I am trying to find State Transition Matrix (STM) in MATLAB using "expm(A*t)" command. For real roots, its fine, but when its complex eigenvalues, its going into piecewise mode and not giving the STM in terms of sin or cosine.
A = [0 1 0; 0 0 1; -2 -3 -5];
syms t;
A_STM = expm(A*t)
For this I am getting following STM in truncated form as attached (One Real and Two Complex Conjugate):
But I want answer in the following form (Not for above example), from MATLAB - Can somebody guide, help?
Thank you.

Star Strider on 23 Mar 2021
Use the rewrite function, then simplify:
A = [0 1; -2 -2];
syms t;
A_STM = expm(A*t)
A_STM = rewrite(A_STM, 'sincos')
A_STM = simplify(A_STM, 500)
producing:
A_STM =
[2^(1/2)*exp(-t)*sin(t + pi/4), exp(-t)*sin(t)]
[ -2*exp(-t)*sin(t), 2^(1/2)*exp(-t)*cos(t + pi/4)]
in LaTeX: .
Star Strider on 24 Mar 2021
My pleasure!

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