rsf2csf

Transforms a real Schur form to a complex Schur form.

Syntax

[UM, TM] = rsf2csf(U, T)

Inputs

U
A unitary matrix.
T
A real Schur matrix.

Outputs

UM
A unitary matrix.
TM
A complex Schur matrix.

Examples

Convert the real Schur form to the complex schur form: 
U = [0.4282    0.0425    0.8965   -0.1054;
    0.5192   -0.2047   -0.1421    0.8175;
    0.7309   -0.0320   -0.4114   -0.5436;
    0.1140    0.9774   -0.0822    0.1580];
    
T = [4.4522    1.6545   -0.5498    0.3187;
         0    0.5742   -2.3726    0.4390;
         0    1.1464    0.5742    0.5319;
         0         0         0    1.3994];

[UM, TM] = rsf2csf(U,T)
UM = [Matrix] 4 x 4
0.42820 + 0.00000i   0.51169 + 0.03490i  -0.02426 - 0.73613i  -0.10540 + 0.00000i
0.51920 + 0.00000i  -0.08111 - 0.16808i   0.11684 + 0.11668i   0.81750 + 0.00000i
0.73090 + 0.00000i  -0.23481 - 0.02628i   0.01826 + 0.33781i  -0.54360 + 0.00000i
0.11400 + 0.00000i  -0.04692 + 0.80256i  -0.55787 + 0.06750i   0.15800 + 0.00000i
TM = [Matrix] 4 x 4
4.45220 + 0.00000i  -0.31381 + 1.35853i  -0.94433 + 0.45145i   0.31870 + 0.00000i
0.00000 + 0.00000i   0.57420 + 1.64923i   1.22620 + 0.00000i   0.30359 - 0.36047i
0.00000 + 0.00000i   0.00000 + 0.00000i   0.57420 - 1.64923i  -0.25057 + 0.43675i
0.00000 + 0.00000i   0.00000 + 0.00000i   0.00000 + 0.00000i   1.39940 + 0.00000i

Comments

When the real Schur form has an upper quasi-triangular matrix T, it indicates the the presence of complex Eigenvalues. The complex Schur form has an upper triangular matrix TM with the complex Eigenvalues on the diagonal of TM.

A = U * T * U' = UM * TM * UM', where U and UM are both unitary.

If the matrix has only real Eigenvalues, then the two forms are the same.